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Shadowed triplet pairings in Hund’s metals with spin-orbit coupling

Jonathan Clepkens,1 Austin W. Lindquist,1 and Hae-Young Kee1, 2, ∗

1Department of Physics and Center for Quantum Materials,

University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada2Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada

Hund’s coupling in multiorbital systems allows for the possibility of even-parity orbital-antisymmetric spin-triplet pairing, which can be stabilized by spin-orbit coupling (SOC). Whilethis pairing expressed in the orbital basis is uniform and spin-triplet, it appears in the band basis asa pseudospin-singlet, with the momentum dependence determined by the SOC and the underlyingtriplet character remaining in the form of interband pairing active away from the Fermi energy.Here, we examine the role of momentum-dependent SOC in generating nontrivial pairing symme-tries, as well as the hidden triplet nature associated with this interorbital pairing, which we dub a“shadowed triplet”. Applying this concept to Sr2RuO4, we first derive several forms of SOC withd-wave form-factors from a microscopic model, and subsequently we show that for a range of SOCparameters, a pairing state with s+ idxy symmetry can be stabilized. Such a pairing state is distinctfrom pure spin-singlet and -triplet pairings due to its unique character of pseudospin-energy lock-ing. We discuss experimental probes to differentiate the shadowed triplet pairing from conventionalpseudospin-triplet and -singlet pairings.

I. INTRODUCTION

A key feature of superconductivity (SC) is theantisymmetric-wavefunction of the Cooper pair under theexchange of two electrons. This has limited the focus toeven-parity spin-singlet and odd-parity spin-triplet pair-ings in a single Fermi surface (FS) system. In multior-bital systems, additional possibilities arise due to the or-bital degree of freedom, such as even-parity spin-triplet,or odd-parity spin-singlet pairings. For example, it wasshown earlier that Hund’s coupling in multiorbital sys-tems allows for an even-parity spin-triplet pairing that isorbitally antisymmetric between two orbitals [1–7]. TheHund’s coupling, significant in transition metal systems,acts as an attractive pairing interaction [1]. Similar to theattractive Hubbard model, the Hund’s coupling allows astrong local Cooper pair to form between two orbitalswith spin-triplet character.However, when the electron motion is introduced, i.e.,

the kinetic term in the Hamiltonian, the pairing is dras-tically weakened, because the electrons comprising theCooper pair with momenta k and −k in different or-bitals have different energies, due to the different elec-tronic dispersions of the orbitals. Thus, to stabilize suchan orbital antisymmetric pairing, significant orbital de-generacy is required throughout momentum space nearthe FS [1, 2]. Furthermore, hybridization between differ-ent orbitals can further weaken interorbital pairings, bysplitting the degeneracy where it occurs.The story becomes more interesting when spin-orbit

coupling (SOC) is introduced. Note that with SOC,purely spin-singlet and -triplet pairings are no longerwell-defined, since the spin and orbital degrees of free-dom are coupled. Thus one defines Cooper pairs in the

∗ hykee@physics.utoronto.ca

total angular momentum (pseudospin) basis. When theSOC is strong, this forms the basis for SC in the heavyfermion superconductors, such as UPt3, leading to odd-parity pseudospin-triplet SC [8, 9]. When the SOC isintermediate, i.e., comparable to the orbital degeneracysplitting terms, spin and orbital characters vary continu-ously, and one can define the pairing in either the orbitalor band basis. Spin-orbital mixing then helps to stabilizethe interorbital pairing. It was shown that SOC indeedenhances even-parity orbital-antisymmetric spin-tripletpairing [3]. The SOC not only supports the orbital-antisymmetric spin-triplet pairing, but it also transformspure spin-triplet pairing into “both” pseudospin-singletand -triplet pairings in the band basis.

Furthermore, the form of the SOC is not limited toatomic SOC, i.e., Li · Si, where i is the site index, andthe precise form can determine the momentum depen-dence of the superconducting state. For example, s-waveSC proximate to a topological insulator or strong RashbaSOC with broken inversion symmetry leads to an effec-tive p+ ip SC, which is odd-parity and a spinless triplet[10–12]. For materials with inversion symmetry, there isstill the possibility of even-parity momentum-dependentSOC [13] (k-SOC), which has been discussed in the con-text of the unconventional superconductor, Sr2RuO4 [14–20]. Similar to a Rashba-SOC generated p + ip SC, theinclusion of k-SOC in a microscopic model with s-wavepairing is reflected in an intraband pairing with the samemomentum character as the SOC.

Here we study a microscopic route to k-SOC and howSOC transforms even-parity interorbital spin-triplet SCinto pseudospin-singlet and -triplet SC in a Hund’s metal,where the multiorbital nature and strong Hund’s cou-pling are crucial. We also illustrate how pseudospin-triplet interband pairing remains, dubbed a “shadowedtriplet” away from the Fermi energy. While this SC be-haves like a singlet in response to low-energy excitations,its hidden identity shows up at finite magnetic fields, and

http://arxiv.org/abs/2009.08597v2mailto:hykee@physics.utoronto.ca

2

it can be tested when the field strength reaches an ap-preciable percentage of the superconducting gap size [21].Applying the concept to Sr2RuO4, we show that s+ idxypairing is stabilized in the t2g orbitals.

The paper is organized as follows. We first introducethe generic Hamiltonian that we will be concerned withthroughout in Sec. II. We then consider a simple butgeneral two-orbital model to show how even-parity spin-triplet pairing arises in Sec. III. This includes the stabil-ity conditions and how SOC transforms this pairing intoa pseudospin-singlet and -triplet in the Bloch band ba-sis. The SOC in the shadowed triplet not only plays anessential role in enhancing the pairing, but it also deter-mines the pairing symmetry. In Sec. IV we investigatemicroscopic routes to several k-SOC terms with d-wavesymmetry, which can lead to various d-wave pairing sym-metries on the FS. In Sec. V, we apply the shadowedtriplet pairing scenario to the prominent unconventionalsuperconductor, Sr2RuO4 [14–17], for which the SOC hasbeen shown to be important [3, 22–26], and we discussthe leading instability towards s+ idxy pairing within athree-orbital model for a range of SOC parameters.

II. GENERAL MICROSCOPIC HAMILTONIAN

We first introduce the generic Hamiltonian that wewill be considering throughout. The Hamiltonian con-sists of three terms. The kinetic term, H0, denotes atight-binding (TB) model, for which we will discuss theprecise form in the subsequent sections. The SOC Hamil-tonian, HSOC , refers to the atomic SOC written in thebasis of t2g orbitals and various momentum-dependentterms, also specified later. The interaction, Hint, has theform of the Kanamori interactions,

Hint =U

2

∑

i,a,σ 6=σ′na,iσna,iσ′ +

U ′

2

∑

i,a 6=b,σσ′na,iσnb,iσ′

+JH2

∑

i,a 6=b,σσ′c†a,iσc

†b,iσ′ca,iσ′cb,iσ

+JH2

∑

i,a 6=b,σ 6=σ′c†a,iσc

†a,iσ′cb,iσ′cb,iσ,

(1)where U and U ′ are the intra- and interorbital Hubbardrepulsions, JH is the Hund’s coupling and c

†a,iσ is an

electron operator creating an electron at site i in orbitala with spin σ. Decoupling these interaction terms intoeven-parity zero-momentum spin-singlet and spin-triplet

order parameters[3, 7, 20, 21] gives,

Hint =4U

N

∑

a,kk′

∆̂s†a,k∆̂sa,k′

+2(U ′ − JH)

N

∑

{a 6=b},kk′d̂†a/b,k · d̂a/b,k′

+4JHN

∑

a 6=b,kk′∆̂s†a,k∆̂

sb,k′

+2(U ′ + JH)

N

∑

a 6=b,kk′∆̂s†a/b,k∆̂

sa/b,k′ ,

(2)

where N is the number of sites, and the spin-triplet and-singlet order parameters are defined as

d̂a/b,k =1

4

∑

σσ′

[iσyσ]σσ′(

ca,kσcb,−kσ′ − cb,kσca,−kσ′)

∆̂sa/b,k =1

4

∑

σσ′

[iσy]σσ′(

ca,kσcb,−kσ′ + cb,kσca,−kσ′)

∆̂sa,k =1

4

∑

σσ′

[iσy]σσ′ca,kσca,−kσ′ ,

(3)and {a 6= b} represents a sum over the unique pairs oforbital indices. An attractive interorbital spin-triplet,

d̂a/b,k, channel is present when JH > U′, which will be

our focus. While this corresponds to a Hund’s couplinglarger than the typical value of ∼ 0.2U for Sr2RuO4[26, 27], this type of pairing instability has also beenfound in several studies beyond mean field (MF) the-ory without such a requirement[4–6]. Furthermore, we

note that the interorbital, ∆̂sa/b,k, and intraorbital, ∆̂sa,k,

spin-singlet order parameters appear with repulsive inter-actions, but they can be induced by the spin-triplet or-der parameters through the SOC. Indeed, calculating thespin-singlet order parameters within a MF spin-tripletpairing state shows that they are generally one order ofmagnitude smaller than the spin-triplet order parame-ters.

III. TWO-ORBITAL MODEL

To illustrate how the shadowed triplet pairing occurs,we first consider a two-orbital MF Hamiltonian, HMF ,consisting of a generic TB model with a specific SOCand an even-parity spin-triplet pairing term. The fol-lowing model allows for a systematic study of the mi-croscopic components relevant to such pairing in multi-orbital systems and is generalized to systems with three

orbitals. We use Ψ†k= (ψ†

k, T ψT

kT −1), where T indicates

time-reversal and ψ†k= (c†a,k↑, c

†b,k↑, c

†a,k↓, c

†b,k↓) consists

of electron operators creating an electron in one of thetwo orbitals a, b with spin σ =↑, ↓. We also introduce thePauli matrices plus identity matrix, ρi, σi, τi, (i = 0, ...3),in the particle-hole, spin and orbital spaces respectively,

3

0 π0

π(a)

tab

(0 − 0.1, 0, 0)

0 π0

π(b)

∆/∆max

(tab, t−, λ) =

t−

(0.1, 0 − 0.13, 0)

0 π0

π(c)

λ0

(0.1, 0.13, 0 − 0.48)

0 π0

π(d)

λd

(0.1, 0.13, 0 − 0.48)

−1.0

−0.5

0.0

0.5

1.0

FIG. 1. Depiction of the gap on the FS in the upper quarter of the first Brillouin zone (BZ) in the two-orbital model, with thearrows indicating the direction the contours move in as the respective parameter is increased and the dashed lines correspondingto the underlying FS. All parameters are given in units of 2t1; see the main text for other parameter details. (a) The SOC andorbital polarization are set to zero while the strength of the orbital hybridization, tab in tk = −4tab sin kx sin ky, is increasedfrom zero to 0.1. This shows that the pairing is decreased by the orbital hybridization tab. (b) tab fixed at 0.1, and the strengthof the orbital polarization, t− in ξ−k = 2t

−(cos kx− cos ky) is increased from zero to 0.13, resulting in zero gap everywhere. Theorbital polarization t− further weakens the pairing. Part (c) shows the revival of the gap for tab = 0.1, t

− = 0.13, as the atomicSOC λ0 is increased from zero to 0.48 demonstrating that SOC drastically enhances the pairing. Part (d) shows the same as(c) but with the d-wave SOC parameter λd in λk = λd(cos kx − cos ky), instead of λ0, increased from zero to 0.48. SOC notonly enhances the pairing, but also determines the momentum dependence.

where the direct product between them is implied. HMFis given by,

HMF =∑

k

Ψ†k

(

H0(k) +HzSOC(k) +Hpair(k)

)

Ψk

H0(k) = ρ3(ξ+

k

2σ0τ0 +

ξ−k

2σ0τ3 + tkσ0τ1

)

HzSOC(k) = −λkρ3σ3τ2Hpair = −dza/bρ2σ3τ2,

(4)

where the TB contribution is given by the sum of thetwo orbital dispersions, ξ+

k, and the difference between

them, ξ−k, which are defined by ξ±

k= ξa

k± ξb

k, as well

as the orbital hybridization of the two orbitals, tk. Herewe consider HzSOC , with the form λkLzSz, to illustratethe effects of the SOC. Later, we derive several possiblek-SOC from a microscopic perspective in Sec. IV, whichdiffer from the LzSz form included here. The pairingHamiltonian is obtained through a MF decoupling ofHintand favors the even-parity interorbital spin-triplet orderparameter via HzSOC , with the d-vector aligned along thez-direction, and thus dza/b is given by

dza/b ≡ (U ′ − JH)1

2N

∑

k

〈d̂za/b,k〉. (5)

To understand the stability of the SC state, we con-sider the relationship between the various components ofour model and their effects on the quasi-particle (QP)dispersion. The gap on the FS is shown in Fig. 1 forvarious cases of ξ−

k, tk and λk. Here, ξ

ak= −2t1 cos ky −

2t2 cos kx−µ, and for ξbk we take x↔ y. The orbitals arecoupled through the SOC, for which we take two cases:the atomic SOC denoted by λ0, and a d-wave SOC givenby λk = λd(cos kx − cos ky), as well as the orbital hy-bridization, which we take as tk = −4tab sinkx sin ky.With these dispersions, ξ−

k= 2t−(cos kx − cos ky), where

t− = t1 − t2 and all parameters are given in units of2t1 = 1. The gap over the FS is shown for four cases:(a) λ = 0, ξ−

k= 0, as tab is increased from zero to 0.1;

(b) keeping λ = 0, with tab = 0.1 as ξ−k

is increased bytuning t− from zero to 0.13; (c) both tab and t− are keptthe same as λ0 is increased from zero to 0.48; and (d)the same as (c) but with the d-wave SOC, instead of λ0,increased by tuning λd from zero to 0.48.When the orbital dispersions are completely degener-

ate, we see that the gap is non-zero everywhere over thetwo identical bands, as shown by the middle contour inFig. 1(a). As the strength of the hybridization, tk, isincreased from zero, the energy separation of the two or-bital dispersions increases wherever tk 6= 0 and the gaparising from interband pairing disappears on the FS, ex-cept where tk vanishes along the kx/y = 0 lines, where thetwo orbital dispersions remain degenerate. Starting fromthere with zero interband pairing over most of the FS,Fig. 1(b) demonstrates the effect of ξ−

kin further reducing

the interband pairing to zero everywhere on the FS, dueto the absence of phase space for zero-momentum pair-ing. Keeping the same parameters used in (b), Fig. 1(c)reveals how the SOC revives the SC state by allowing foran intraband pairing on the FS. As the SOC is increasedfrom zero, the intraband gap becomes non-zero over theentire FS. Additionally, the sign of the gap function isopposite on the two bands, matching the ∆s(k)τ̃3 depen-

4

dence, which we will show below in Eq. (10) and Eq. (11),but uniform on each band due to the lack of momentumdependence of the atomic SOC. In contrast, Fig. 1(d)displays the d-wave dependence of the gap arising fromthe d-wave SOC. Thus with the introduction of tk, ξ

−k

and λk, the pairing on the FS is transformed from aninterband spin-triplet to a purely intraband pseudospin-singlet with the same momentum dependence as the as-sociated SOC, while the pseudospin-triplet is active awayfrom the Fermi energy.

The above results can also be understood via thecommutation relations between the order parameter andother terms. The pair-breaking effects are revealed bythe commuting behavior with the pairing term [28–30].Conversely, the SOC anti-commutes with the pairingterm and generally enhances the pairing state. The anti-commutators are given by

{ξ−k

2ρ3σ0τ3, Hpair} = ξ−k dza/bρ1σ3τ1

{tkρ3σ0τ1, Hpair} = −2tkdza/bρ1σ3τ3{−λkρ3σ3τ2, Hpair} = 0.

(6)

These effects are reflected in the QP dispersion, given by

Ek = ±1

2

[

ξ−2k

+ ξ+2k

+ 4[

t2k+ (dza/b)

2 + λ2k

]

± 2√

ξ+2k

[

ξ−2k

+ 4t2k+ 4λ2

k

]

+ 4(dza/b)2[

ξ−2k

+ 4t2k

]

]1

2

.

(7)The general equation of the FS is ξ−2

k= ξ+2

k−4(t2

k+λ2

k),

from which it can be seen that if λk and tk = 0, andthe orbitals are degenerate, i.e., ξ−

k= 0, we recover the

conventional BCS result for the gap energy on the FS:±dza/b. In the general case where these terms are non-zero, assuming dza/b is small, the QP gap is

QP gapon FS ≈ ±√

(dza/b)4 + 16λ2

k(dza/b)

2

4(ξ−2k

+ 4(t2k+ λ2

k)). (8)

From this, it is clear that increasing ξ−k

and tk decreasesthe overall gap size. The detrimental effects on the gap

size, signified by the commuting nature of both tk andξ−k

with the pairing Hamiltonian, are a result of shiftingapart in energy the bands being paired, resulting in thegap moving away from the FS. However, turning on theSOC significantly enhances the gap size [3], as shownin Fig. 1(c), and as the SOC strength becomes largerthan ξ−

kand tk, the gap can be restored to the order of

dza/b. Furthermore, if λk has a d-wave form factor such as

(cos kx − cos ky), the gap reflects the exact same d-wavesymmetry, as shown in Fig. 1(d). The enhancement ofthe SC state is accomplished by providing a non-zerointraband pseudospin-singlet pairing on the FS, as weshow below in the band basis.With the aim to further understand the nature of the

SC state, we study how the pairing transforms to theBloch band basis, labeled by band indices α, β and pseu-dospin s = (+,−). The transformation is given by(

ca,kσ

cb,kσ

)

=

(

ησ+12 fk −

ησ−12 f

∗k

−gkgk

ησ+12 f

∗k− ησ−12 fk

)(

αk,s

βk,s

)

(9)where ησ = +1(−1) for σ =↑ (↓) and s =+(−). The coefficients of the transformation aregiven by fk = − γk|γk|

√

12 (1 +

ξ−k√

ξ−2k

+4|γk|2) and gk =

−√

12 (1 −

ξ−k√

ξ−2k

+4|γk|2), where fk is chosen to be com-

plex with the same phase as γk = tk + iλk, gk is real,and |fk|2 + g2k = 1. Applying this transformation onHpair results in the pairing in the band basis, wherewe have also defined the Pauli matrices ρ̃i, σ̃i, τ̃i in theNambu, pseudospin, and band spaces with the basis

Φ†k= (φ†

k, T φT

kT −1) and φ†

k= (α†

k+, β†k+, α

†k−, β

†k−). We

obtain,

H̃pair(k) = ρ̃2σ̃0[

−∆s(k)τ̃3 −∆sα/β(k)τ̃1]

− dzα/β(k)ρ̃2σ̃3τ̃2,(10)

where ∆s(k) and ∆sα/β(k) denote pseudospin-singlet in-

traband and interband pairings respectively, and dzα/β(k)

is a pseudospin-triplet interband pairing. The intrabandand interband nature of these pairings becomes more ap-parent from the operator form,

H̃pair(k) = i∆s(k)

[

(α†k,+α

†−k,− − α

†k,−α

†−k,+)− (β

†k,+β

†−k,− − β

†k,−β

†−k,+)

]

+i∆sα/β(k)[

(α†k,+β

†−k,− − α

†k,−β

†−k,+) + (β

†k,+α

†−k,− − β

†k,−α

†−k,+)

]

+dzα/β(k)[

(α†k,+β

†−k,− + α

†k,−β

†−k,+)− (β

†k,+α

†−k,− + β

†k,−α

†−k,+)

]

.

(11)

The above equation shows the sign change in the in-traband pairing between the two bands, as displayedin Fig. 1(c) and 1(d). This relative sign between the

bands is similar to the s+− gap structure [7], although itshould be noted that here for simplicity we have ignoredthe SOC-induced intraorbital singlets [3, 7, 21, 31] that

5

would add to the gap on each band. For small (JH −U ′)they are small compared to dza/b and they do not affect

the conclusions of this section. The pairings in the bandbasis have the following form:

∆s(k) = −2dza/bIm(fk)gk = −2dza/bλk

√

ξ−2k

+ 4(t2k+ λ2

k)

∆sα/β(k) = −dza/bIm(f2k) = −2dza/b|fk|2tkλkt2k+ λ2

k

dzα/β(k) = dza/b[g

2k +Re(f

2k)] = d

za/b

(

g2k + |fk|2t2k− λ2

k

t2k+ λ2

k

)

.

(12)While the orbital order parameter is s-wave and containsno explicit momentum dependence, transforming to theband basis generates potentially complex momentum de-pendence from SOC and orbital hybridization. The or-bital spin-triplet order parameter carries over to the in-terband pseudospin-triplet, which, in the limit of zeroSOC, becomes equal to dza/b, while both the intraband

and interband pseudospin-singlets vanish. The interbandpseudospin-singlet pairing also vanishes for tk = 0. How-ever, an important feature for λk 6= 0 is the presence ofthe intraband pseudospin-singlet pairing, ∆s(k), whichacquires the same symmetry dependence as a function ofmomentum as λk, as shown in Fig. 1(d). While the inter-band pseudospin-triplet is a signature of the fundamentalinterorbital spin-triplet order parameter, it is the intra-band pseudospin-singlet that leads to a weak-couplinginstability. This is because the interband pairing contri-bution to the gap will generally be negligible on the FS,such that the gap is given by |∆s|. Considering the QPdispersion in terms of the band pairings,

Ek = ±[

(ξαk)2 + (ξβ

k)2

2+ (∆s)2 + (∆sα/β)

2 + (dzα/β)2

±

√

[

(ξαk)2 − (ξβ

k)2]2

4+ (ξα

k− ξβ

k)2[

(∆sα/β)2 + (dzα/β)

2]

]1

2

,

(13)and evaluating this on either the α or β FS, for only∆s intraband pairing, we obtain the gap energy ±|∆s|.For either only ∆sα/β or d

zα/β interband pairing, a gap

of ±∆sα/β or ±dzα/β forms where ξαk = −ξβk. This

corresponds to an energy gap on the FS only where

ξαk

= ξβk= 0, which is not a generic feature but rather

requires fine-tuning to achieve, otherwise the gap formedwill be away from the Fermi energy.While the model introduced in this section is simple,

its generality gives insight into the role of the orbitaldegeneracy, hybridization, and SOC for multiorbital sys-tems in dictating the stability of the even-parity spin-triplet SC state. It is straightforward to extend to three-orbital descriptions. Furthermore, we have seen that inthe band basis, the intraband pairing on the FS takeson the momentum dependence of the SOC, allowing for

a rich collection of pairing symmetries unexpected fromthe original s-wave order parameter and in contrast toother forms of momentum-dependent SC that arise fromnonlocal interactions. However, the possible pairing sym-metries will depend on the forms of k-SOC that can beobtained from microscopic considerations. Therefore, wenow turn to a study of how various forms of k-SOC canarise microscopically.

IV. MICROSCOPIC ROUTE TO

MOMENTUM-DEPENDENT SOC

Here, we take as a specific microscopic example thelayered perovskite Sr2RuO4, which has the tetragonalspace group I4/mmm and point group D4h, for whichthe Ru 4d t2g orbitals are the relevant low-energy de-grees of freedom. With this, we study how the variousforms of k-SOC with different d-wave form factors suchas (a) an in-plane dxy SOC in the B2g representation, (b)in-plane dx2−y2 SOC in the B1g representation, and (c)interlayer {dxz, dyz} SOC in the Eg representation canarise microscopically, going beyond a purely symmetry-based analysis.

A. In-plane B2g

We begin by discussing the in-plane k-SOC in theB2g representation, which contains (sin kx sin ky) momen-tum dependence. This SOC is important since based on

symmetry, there will already be a finite HB2gSOC in the

presence of the orbital hybridization and atomic SOC.Since the orbital hybridization for the system consid-ered here will appear between the dxz and dyz orbitals

as Hab = −4tab∑

kσ

sin kx sinky(

c†yz,kσcxz,kσ + H.c.)

,

and transforms under the B2g representation, there isa cubic coupling term in the free energy between theatomic SOC (λ), transforming as A1g, orbital hybridiza-tion and B2g SOC. This symmetry-allowed coupling ∼〈HB2gSOC〉〈H

A1gSOC〉〈Hab〉, where H

A1gSOC denotes the atomic

SOC, ensures the presence of a non-zero B2g SOC in thepresence of both orbital hybridization and atomic SOC.Furthermore, the in-plane k-SOC in the B2g represen-

tation arises through several hopping channels, all uti-lizing intermediate p-orbitals, but different oxygen sitesdenoted by 1-5 in Fig. 2, including only nearby sites. Asmost of these channels occur in a single layer of Ru-O oc-tahedra, this SOC should be expected to be the leadingcontribution beyond the atomic SOC, since we will seethat the other two k-SOC always require hopping to ad-ditional layers. Such an in-plane k-SOC can be obtainedthrough perturbation theory by considering hopping be-tween next-nearest neighbor Ru atoms through the vari-ous channels, via the oxygen sites as intermediate states,and including the oxygen p-orbitals’ atomic SOC. Thishopping process results in an electron hopping from the

6

FIG. 2. Hopping channels generating the in-plane B2g d-wavek-SOC for the dxz and dxy orbitals (green and red), whichconsists of an effective spin-flip hopping between next-nearestneighbor sites. The alternative process involving the dyz or-bital is related by a C4 rotation and only the interlayer pro-cess is shown in detail for clarity. (a) The intermediate oxy-gen sites for the contributing hopping channels are indicatedby the numbering (1-5), with the relevant p orbitals (yellowand blue) and the intermediate hopping amplitudes shown bydashed lines only for the fifth channel. The other channelsinvolve the same p-orbitals at sites 1-4. The top layer of theunit cell is also removed for clarity. (b) Schematic top view ofthe hopping process, where the bottom lobe of the pz orbitalis shown.

dxy orbital with spin σ to either a dxz or dyz orbital withspin −σ, where the former case is shown in Fig. 2 by theblack solid line. The contributing channels are indicatedby the numbering of the intermediate oxygen sites, andonly path 5 is shown explicitly for clarity. The effectiveSOC Hamiltonian involving the next-nearest neighbor Rusites is obtained by

HB2gSOC =

∑

p±

H0|p±〉〈p±|H0Ed − Ep±

, (14)

where H0 denotes the hopping Hamiltonian involvingboth d and p orbitals. The sum runs over the intermedi-ate oxygen states for all channels, which are eigenstatesof the oxygen SOC |j,mj , r〉, with r the position of theoxygen site, and we consider up to the 2nd order of theperturbation theory for this process.For instance, considering a hopping process between

a dxy state with spin-↑ and a spin-↓ dxz state via theapical oxygen site shown, we have |p+〉 = − 1√6 (|px, ↓〉 + i|py, ↓〉) +

√2√3|pz, ↑〉 and |p−〉 = − 1√3 (|px, ↓〉 + i|py, ↓

〉)− 1√3|pz, ↑〉. The energy denominator is given for |p+〉

and |p−〉 by Epd + λp2 and Epd − λp respectively, whereEpd is the difference in the on-site atomic potentials, andλp is the oxygen SOC constant. Considering the hoppingamplitude between the dxy orbital with spin σ at site Rdenoted by |xy, σ,R〉 and the opposite spin state of the

dxz orbital at site R′, |xz,−σ,R′〉, we obtain,

〈xz,−σ,R′|HB2gSOC |xy, σ,R〉 =

ησλp

(Epd +λp2 )(Epd − λp)

∑

i

taipd,itbipd,i,

(15)

where the sum is over the contributing channels involv-ing the different oxygen sites indicated in Fig. 2 and ai, birefer to the orbitals in the intermediate hopping ampli-tudes. All of these channels contain the hopping ampli-tudes between either the dxy/px and dxz/pz, or dxy/pzand dxz/px orbitals. From Fig. 2 it can be seen thatthe overall sign dependence of the hopping channel willmatch that of the dxy orbital, since px and dxz changesign identically under the yz and xz mirror planes, whilepz is even under them, leading to the sin kx sin ky depen-dence. Furthermore, the presence of ησ in Eq. (15) givesrise to σy spin-dependence.Taking into account the equivalent process between the

dxy and dyz orbitals, which is related by a C4 rotation,we obtain

HB2gSOC = 4iλ

B2g∑

kσσ′

sinkx sin kyσyσσ′c

†xz,kσcxy,kσ′

− 4iλB2g∑

kσσ′

sin kx sin kyσxσσ′c

†yz,kσcxy,kσ′ +H.c.,

(16)

where λB2g =λp

(Epd+λp

2)(Epd−λp)

∑

i taipd,it

bipd,i and i =

1, .., 5 indicates the different p-orbital sites involved inFig. 2.Quantifying the value of λB2g in Sr2RuO4 will require

further studies to accurately include the additional ef-fects of the coupling between other microscopic param-eters such as the Ru on-site SOC and orbital hybridiza-tion, as discussed above. Correlation effects have alsobeen shown to be crucial for the size of SOC, whichis enhanced from the local-density approximation SOCvalues [26]. The current work is to show a microscopicroute to generating k-SOC within a perturbation theoryapproach, beyond a purely symmetry-based perspective.These considerations also apply to the other k-SOC pro-cesses, to which we now turn.

B. In-plane B1g

We now consider a microscopic route to obtaining anin-plane k-SOC in the B1g representation, which has adx2−y2 form factor. This requires a different layer of Ru-O octahedra, but the same procedure. An example ofthis process is shown in Fig. 3(a), for which the hop-ping between the dxy orbital with a spin σ state andthe opposite spin state of the dyz orbital is indicated,where the overlap of the dxy orbital with both the pxand py orbitals is shown. The hopping amplitudes arerepresented schematically in Fig. 3(b) for both the +x̂or +ŷ directions, where the pz lobe closest to the plane

7

FIG. 3. Hopping processes generating the in-plane B1g k-SOC, which consists of an effective spin-flip hopping betweennearest neighbor sites indicated by the solid line. (a) The rel-evant d and p orbitals shown in the three-dimensional (3D)structure and the dashed lines indicating the intermediatehopping processes. The process involving the dyz orbital isshown as an example, with the alternative process involvingthe dxz orbital related by a C4 rotation. The px and py or-bitals are separated for clarity. (b) Schematic picture of thehopping process from a top view where only the lobe of thepz wavefunction closest to the plane of the hopping is shownand the numbers indicate whether the orbital is on the apicaloxygen site (2) or bottom (1) layer. The px and py orbitalsare also shown in two separate squares for clarity.

containing the effective hopping is shown, and the px/pyorbitals are in separate squares for clarity. The number-ing in Fig. 3(b) indicates whether the orbital is on theapical oxygen site (2) or the bottom (1) layer. We de-note the hopping between the dxy orbital and px(py) astxy,xpd (t

xy,ypd ), with both having magnitude t

xypd due to the

C4 rotational symmetry. The hopping between either ofthe dyz/dxz orbitals and pz is denoted by t

zpd, where the

two are also of equal magnitude.

Evaluating the sum over the possible intermediatestates for a given apical oxygen site, we obtain

〈yz,−σ,R′|HB1gSOC |xy, σ,R〉 =

− λp2(Epd +

λp2 )(Epd − λp)

∑

r

tzpd

[

ησtxy,xpd + it

xy,ypd

]

.

(17)Due to the mirror symmetry about the xz plane, wherethe apical oxygen above the plane in the +x̂ + ŷ di-rection is shown in Fig. 3, the imaginary term cancelsafter summing over the two oxygen sites at ±ŷ, sincetzpdt

xy,ypd → −tzpdt

xy,ypd , while the real term is invariant.

FIG. 4. Hopping process generating the Eg k-SOC and in-terlayer hopping. (a) The relevant d and p orbitals shown inthe 3D structure, where the dashed lines indicate the interme-diate hopping processes. Only the process involving the dxzorbital hopping in the −ẑ direction is shown for clarity. (b)A schematic picture of the hopping process from a top viewwhere the bottom lobes of the dxz orbital are shown on top ofthe px orbital and the numbers indicate whether the orbitalis on the top (3), bottom (1) layer or the apical oxygen site(2).

Summing over both x- and y-direction paths, we have,

HB1gSOC = 2iλ

B1g∑

kσσ′

σyσσ′ (cos kx − cos ky)c†yz,kσcxy,kσ′

+ 2iλB1g∑

kσσ′

σxσσ′ (cos kx − cos ky)c†xz,kσcxy,kσ′

+H.c.,(18)

where λB1g = 2txypd

tzpdλp

(Epd+λp

2)(Epd−λp)

and it can be noted that

this Hamiltonian is similar to λ(LxSx −LySy), but withλ replaced with a dx2−y2 form factor.

C. Interlayer Eg

Here we consider an interlayer k-SOC as well as inter-layer hopping between the dxy and either of the dxz/dyzorbitals. The k-SOC is the spin-dependent part of thisprocess that occurs between states with the same spinin adjacent layers, via the apical oxygen sites as shownin Fig. 4(a), which displays the hopping process for thedxz/dxy case. A schematic illustration of the relevanthopping amplitudes is represented in Fig. 4(b), wheretxy,xpd (t

xy,ypd ) is the hopping amplitude between the dxy and

px(py) orbitals in different layers with equal magnitudes,txypd . The hopping amplitude occurring purely in the ẑ di-rection between the dxz orbital and the px at the apicaloxygen site is txz,xpd .

8

We have for the matrix element represented by Fig. 4,

〈xy, σ,R′|HEgSOC +HEglayer |xz, σ,R〉 =

txz,xpd

2(Epd +λp2 )(Epd − λp)

[

txy,xpd (2Epd − λp)− iησλptxy,ypd

]

,

(19)from which we see that the real part gives a spin-independent hopping between the orbitals and the imag-inary part gives a spin-dependent hopping. Both realand imaginary parts are odd in z since txz,xpd is odd withrespect to reflection about the xy plane. Furthermore,since the real and imaginary parts are proportional totxy,xpd and t

xy,ypd respectively, the real part will be even

(odd) in x(y) with the opposite signs for the imaginarypart. A C4 rotation yields the equivalent result involvingdyz, but with an opposite even/odd sign dependence withrespect to the x, y directions. The result is an effectiveinterlayer hopping,

HEglayer = −8tz

∑

kσ

coskx2

sinky2

sinkz2c†xy,kσcxz,kσ

− 8tz∑

kσ

sinkx2

cosky2

sinkz2c†xy,kσcyz,kσ +H.c.,

(20)

as well as the effective SOC,

HEgSOC =− 8iλEg

∑

kσσ′

σzσσ′ sinkx2

cosky2

sinkz2c†xy,kσcxz,kσ′

+8iλEg∑

kσσ′

σzσσ′ coskx2

sinky2

sinkz2c†xy,kσcyz,kσ′

+H.c.,(21)

where the effective hopping amplitudes are

tz =txypdt

xz,xpd

2(Epd +λp2 )(Epd − λp)

(2Epd − λp),

λEg =−txypdt

xz,xpd

2(Epd +λp2 )(Epd − λp)

λp.

(22)

In summary, we have shown how three different k-SOCterms with distinct symmetries can be generated withina model consisting of the t2g orbitals and oxygen p-orbitals with associated on-site SOC. A fit to the density-functional theory (DFT) results of Ref. [25] was carriedout within a TB model including k-SOC in Ref. [20], forwhich the size of the various k-SOC parameters were alldetermined to be O(1meV). However, as discussed inRef. [20], through comparison to angle-resolved photoe-mission spectroscopy (ARPES) measurements [26], it canbe seen that the DFT parameters do not accurately ac-count for the size of the SOC, which is enhanced throughcorrelation effects [26]. Thus, as mentioned previously,quantifying the values of k-SOC in Sr2RuO4 requires fu-ture studies.

With a microscopic understanding of the origin of thesethree forms of k-SOC, we next turn to incorporatingthem into a three-orbital model and study the pairinginstabilities that arise in Sr2RuO4 when the on-site in-teractions are included.

V. APPLICATION TO Sr2RuO4

We apply the shadowed triplet pairing scenario tothe unconventional superconductor Sr2RuO4 [14–17] byperforming numerical calculations within MF theory forthree t2g orbitals. This includes 18 spin-triplet orderparameters, described in Eq. (3), which are solved self-consistently at zero temperature. Let us take a Hamilto-nian, H = H0 +HSOC +Hpair , where the kinetic term,H0, is given by

H0 =∑

k,σ,a

ǫakc†a,kσca,kσ +

∑

kσ

tkc†yz,kσcxz,kσ +H.c., (23)

with a the orbital index and orbital dispersions,

ǫyz/xzk

= −2t1 cos ky/x − 2t2 cos kx/y − µ1d, ǫxyk =−2t3(cos kx + cos ky) − 4t4 cos kx cos ky − µxy and tk =−4tab sin kx sinky + 4t′ab sin kx2 sin

ky2 cos

kz2 . The TB pa-

rameters, introduced in Ref. [21], are (t1, t2, t3, t4, tab,t′ab, µ1d, µxy) = (0.45, 0.05, 0.5, 0.2, 0.0025, 0.025, 0.54,0.64), where all parameters are in units of 2t3. The SOCHamiltonian is

HSOC = HA1gSOC +H

B2gSOC +H

EgSOC , (24)

where the atomic SOC in the basis of t2g orbitals

is HA1gSOC = iλ

∑

k,abc,σσ′ εabcc†a,kσcb,kσ′σ

cσσ′ , and εabc

is the completely anti-symmetric tensor with a, b, c =(1, 2, 3) = (yz, xz, xy) representing the t2g orbitals. Theresulting TB model is capable of reproducing the ex-perimental FS of Sr2RuO4 [26, 32–34] and the pairingterm, Hpair, consists of the attractive channel expressedin terms of the interorbital spin-triplet order parametersin Eq. (2).The MF results are obtained with JH − U ′ = 0.7 and

the atomic SOC fixed to λ = 0.05. With λB2g = 0, weobtain a purely s-wave solution with dxxz/xy = d

yyz/xy >

dzyz/xz, as shown in Ref. [3]. As λB2g reaches an ap-

preciable percentage of λ, the dxy component becomesnon-zero, with orbital MFs dyxz/xy = d

xyz/xy. Also, there

is a relative phase of π2 compared to the s-component ofpairing. This solution is of the form s+ idxy in the bandbasis and has underlying triplet character with a pre-dominantly in-plane d-vector involving pairing mostly be-tween the dxz and dxy as well as dyz and dxy orbitals. ForλB2g ≈ −0.0305, the dxy and s components are approxi-mately equal in magnitude. The QP gap is found to bemaximum along the kx = ky line, with ∆max = 2.4×10−5and ∆min = 1.1 × 10−6. The s + idxy pairing state ex-ists for a range of λ and λB2g values around this. In-creasing λB2g will eventually lead to a state with only

9

0 π0

π

|∆||max(∆)|

|∆||max(∆)|

dxy|dxy|

0.0

0.2

0.4

0.6

0.8

1.0

−1.0

−0.5

0.0

0.5

1.0

FIG. 5. Gap at the FS at kz = 0 for a representative s+ idxysolution with λ = 0.05, λEg = 0.005 and λB2g = −0.0305, forwhich the maximum gap over the FS is 2.4 × 10−5 and theminimum gap is 1.1× 10−6, where all energies are in units of2t3 (see main text for details). The direction of the arrows in-dicates the in-plane component of the interorbital spin-tripletd-vector associated with the shadowed triplet state, whichtransforms to intraband pseudospin-singlet pairing on the FS.The length of the arrow indicates the magnitude of the in-plane component; the shorter the arrow, the bigger the c-axiscomponent. The inset displays the magnitude of the gap overthe FS throughout the full BZ, with the sign of the dxy com-ponent of the gap function shown.

the dxy component. We note that while we have derived

HB2gSOC through a perturbative process and λ

B2g is there-fore expected to be smaller than the SOC of oxygen, λp ≈20meV [35], there should be additional enhancement ofthe B2g SOC through the coupling to the atomic SOCand orbital hybridization, as well as correlations, whichwe discussed earlier. We therefore treat λB2g as an ef-fective parameter taking these effects into account, whileyielding a FS in agreement with ARPES.We show the gap on the FS in Fig. 5, along with

arrows indicating the nature of the shadowed triplet atselect k points away from the Fermi energy. The SC gapis smallest on the α band along the BZ boundaries, on theorder of 1% of the maximum gap orO(1µeV), using 2t3 ≈700 − 800meV [36, 37], and can be even smaller as theMF gap is generally overestimated. Furthermore, thesedeep minima in the gap are robust to changes in the SOCparameters within the region where the s+ idxy solutionis stabilized. The direction of the arrows indicates the in-plane direction of the d-vector, with the length indicatingthe magnitude of the in-plane component; the shorter thearrow, the bigger the c-axis component. Due to the dxycomponent vanishing along the a- and b- axis, the pairing

solution is composed of dxxz/xy along the a-axis and dyyz/xy

along the b-axis, leading to a d-vector that is parallel tothe respective axis. The vanishing of the dxy componentof the pairing along the kx/ky directions is illustrated inthe inset of Fig. 5.

Within a microscopic theory including Kanamori in-teractions, the atomic SOC and the B2g and Eg k-SOCsderived in Sec. IV, it is possible to stabilize both orderparameters of the s+ idxy or dxz + idyz type at the FS,depending on the relative size of the Eg, B2g and atomicSOC strengths. As discussed in Sec. III, these order pa-rameters will appear as intraband pseudospin-singlets onthe FS, but with underlying interorbital triplet charac-ter originating from the orbital order parameters, da/b.The atomic SOC will stabilize MFs of the form dxxz/xy =

dyyz/xy > dzyz/xz [3, 18, 20, 21]. The B2g SOC, which

is given by Eq. (16), will favor an order parameter thatappears with dxy symmetry and underlying d

xyz/xy and

dyxz/xy triplet character. The Eg SOC, given by Eq. (21),

will favor a multicomponent order parameter that ap-pears as dxz+idyz [18, 20] and have underlying d

zxz/xy and

dzyz/xy character. By including both the atomic and B2gSOCs, a multicomponent order parameter with s+ idxysymmetry can be stabilized, where the relative size ofthe s and dxy components is determined by the relativesizes of λ and λB2g . By comparing the ground-state en-ergies, we find that the dxz+idyz solution becomes favor-able over the purely s-wave solution when λEg ≈ 0.015and λB2g = 0. However, including the B2g SOC byfixing λB2g = −0.0305, the critical value at which thedxz + idyz state is stabilized remains approximately thesame. Given that the only contribution to the Eg SOCinvolves hopping between different layers, it is reason-able that λB2g is larger than λEg , leading to a dominants+ idxy pairing state over dxz + idyz.

There has been a growing body of evidence suggestingthat any viable order parameter must be a time-reversalsymmetry breaking (TRSB) multicomponent order pa-rameter [38–41], with an appropriate symmetry that willlead to a jump in the c66 elastic modulus at Tc [42, 43],and lead to a substantial reduction of the nuclear mag-netic resonance (NMR) Knight shift for an in-plane field[44, 45]. Two even parity proposals are the multicompo-nent {dxz, dyz} order parameter in the two-dimensionalEg representation [20, 46] and the dx2−y2 + igxy(x2−y2)order parameter which relies on an accidental degener-acy with components from both the B1g and A2g rep-resentations [47]. The s + idxy state proposed here isa combination of the A1g and B2g representations andgenerates a sudden change in the shear c66 elastic modu-lus but not (c11 − c12)/2, consistent with the ultrasounddata. This is in contrast to an order parameter in the Egrepresentation, leading to a jump also in the (c11−c12)/2elastic modulus, which has not been observed experi-mentally [42, 43]. Since other pairing solutions such asdyxz/xy = −dxyz/xy are also found as local minima in MFtheory, further experiments and theoretical studies to pin

10

down different order parameters are left for the future.

VI. SUMMARY AND DISCUSSION

In summary, we have studied the microscopic mecha-nisms of k-SOC and its importance for even-parity spin-triplet pairing in Hund’s metals. By taking a simple two-orbital model, we show how a purely interorbital s-wavetriplet pairing in the orbital basis becomes an intrabandpseudospin-singlet pairing with nontrivial momentum de-pendence near the FS, as well as pseudospin-singlet and-triplet interband pairings which also contain momen-tum dependence. In the process, we have illustratedthe effects of orbital hybridization and SOC on the in-terorbital spin-triplet pairing state. Applying the ideato Sr2RuO4, we have derived several forms of d-wave k-SOC in the B1g, B2g and Eg representations, by includingthe SOC of the oxygen sites within a model consisting ofthe t2g orbitals and oxygen 2p orbitals. While determin-ing the precise size of the various k-SOC parameters inSr2RuO4 will require future study, the perturbative ap-proach taken here is an important step in understandingthe microscopic origins of these terms in Sr2RuO4 andother materials. For instance, based on this analysis it isreasonable to expect that the dominant form of k-SOCwill likely be from the next-nearest-neighbor in-plane B2gSOC, which will generate a dxy pairing component inaddition to the s-wave pairing stabilized by the atomicSOC. Subsequently, we have demonstrated the viabilityof the s+ idxy multicomponent solution by including theatomic SOC, B2g and Eg SOCs with t2g orbitals. Fora range of the three SOC parameters, the existence ofthe s+ idxy state with a predominantly in-plane d-vectoris generic and independent of details, while competitionwith other shadowed triplet pairing symmetries dependson the microscopic parameters. The concept we havepresented is also applicable to other correlated Hund’smetals with significant SOC.While the pairing solution we have found manifests as a

pseudospin-singlet on the FS, the shadowed triplet naturewith a predominantly in-plane d-vector will be apparentin the presence of finite fields [48]. As discussed previ-ously in Ref. [21], these properties can be confirmed byNMR under uniaxial strain, with an in-plane field appliedalong both the direction of the strain and perpendicularto it. The s + idxy state we have presented here has es-sentially the same property that near the x/y directions,where there is mostly dxz/dyz and dxy orbital charac-ter, the d-vector is parallel to the crystal axes due to thes-wave component of pairing and the fact that the dxycomponent vanishes along those directions. Therefore,under uniaxial strain, there should be a rotation of theaverage d-vector, leading to an anisotropic Knight shift

between the x and y directions when the field is a signif-icant fraction of the gap size. Interestingly, the low fieldbehavior is governed by the pairing near the FS, leadingto a more isotropic response consistent with the Knightshift of NMR [21].

Going beyond the purely s-wave case, an s+ idxy pair-ing naturally explains experiments suggesting a multi-component order parameter with TRSB and the observedjump in the c66 elastic modulus but not the (c11− c12)/2modulus [38–43]. While dxz + idyz matches the jump inc66, the lack of an observed jump in (c11 − c12)/2 is infavor of the s + idxy pairing state over the dxz + idyzstate. These two pairing states can also be distinguisheddue to their different triplet character. The leading or-der parameter for the dxz + idyz solution corresponds toan out-of-plane d-vector, which would yield no rotationunder uniaxial strain in contrast to the behavior of thes + idxy state, as discussed above. Such an experimentcould provide a test of the s+ idxy order parameter aris-ing from the combination of local interactions and k-SOCin light of the lack of many other viable multicomponentalternatives.

In general, two order parameters from different rep-resentations with broken time-reversal symmetry impliesthe presence of two transition temperatures. However,currently there are conflicting experimental data underuniaxial strain regarding this issue: specific heat mea-surements show no signs of a second transition [49], whilemuon spin relaxation measurements do indicate a split-ting of Tc and the onset of TRSB [41]. Therefore, futureexperiments under strain will be an important test for aTRSB pairing state. Beyond the experiments under uni-axial strain, it will be important going forward for futureexperiments to clarify whether the putative gap nodes[50–55] are indeed nodes or deep gap minima, which arisenaturally in the pairing solution considered here, as wellas the precise location in k-space of these nodes.

ACKNOWLEDGMENTS

We thank Y.-B. Kim for useful discussions. This workwas supported by the Natural Sciences and EngineeringResearch Council of Canada Discovery Grant No. 06089-2016, the Center for Quantum Materials at the Univer-sity of Toronto, the Canadian Institute for Advanced Re-search, and the Canada Research Chairs Program. Com-putations were performed on the Niagara supercomputerat the SciNet HPC Consortium. SciNet is funded by: theCanada Foundation for Innovation under the auspices ofCompute Canada; the Government of Ontario; OntarioResearch Fund - Research Excellence; and the Universityof Toronto.

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