Surface State of Inter-orbital Pairing State in Sr2RuO 4Superconductor Satoshi Ando Satoshi Ikegaya Shun Tamura Yukio Tanaka Keiji Yada Department of Applied Physics Nagoya University Nagoya 464-8603 Japan

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Surface State of Inter-orbital Pairing State in Sr2RuO4Superconductor

Satoshi Ando, Satoshi Ikegaya, Shun Tamura, Yukio Tanaka, Keiji Yada

Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

(Dated: October 17, 2022)

We study the (001) surface state of a recently proposed Egsymmetry inter-orbital-odd spin-

triplet s-wave superconducting (SC) state in Sr2RuO4(SRO). We conﬁrm that this pair potential

is transformed into a chiral d-wave pair potential and a pseudo-Zeeman ﬁeld in the band basis for

a low-energy range. Because of the chiral d-wave pair potential, the surface states appear near zero

energy in the momentum range enclosed by the nodal lines of the chiral d-wave pair potential for

each band at the (001) surface. Nevertheless, the pseudo-Zeeman ﬁeld gives band splitting of the

surface states, and its splitting energy is much smaller than the SC energy gap. The local density of

states (LDOS) at the (001) surface of the SC state has a pronounced peak structure at zero energy

because of the surface states near zero energy when the order of the resolution is lower than the

splitting energy. This peak structure is robust against perturbations, such as an orbital Rashba

coupling or an EuSC pair potential at the surface.

I. INTRODUCTION

The superconducting (SC) symmetry in Sr2RuO4

(SRO) has been a central issue in condensed-matter

physics since its discovery [1]. Spin-triplet chiral p-

wave [kx+ iky-wave] was long considered the leading

candidate for SC symmetry because of the obser-

vations of the constant spin-susceptibility (NMR [2]

and neutron scattering [3]) and time-reversal symme-

try breaking (TRSB) (µSR [4] and Kerr eﬀect [5]).

In addition, numerous theoretical studies have sup-

ported the realization of the kx+ iky-wave state [6–

19]. Recently, however, Pustogow et al. [20] pointed

out that there had been a heating issue in the previ-

ously reported NMR experiment. When this prob-

lem was solved, spin susceptibility under an in-plane

magnetic ﬁeld was suppressed at temperatures be-

low Tcin both NMR and µSR measurements [20–23].

These recent results appear to be inconsistent with

the kx+ iky-wave state. However, ultrasound and

thermodynamics experiments suggest multiple de-

generate order parameters [24–26], and several the-

oretical studies focused on the two-component or-

der parameters with TRSB in SRO have been re-

ported [27–37]. In general, TRSB SC states are

composed of two diﬀerent components that belong

to two diﬀerent irreducible representations (irreps)

or to the same irrep. In cases where two diﬀerent

irreps are mixed, the possibility of certain combina-

tions has been proposed, such as s0+ idx2−y2-wave

[28], dx2−y2+ igxy(x2−y2)-wave [29–32], and s+ idxy-

wave [33, 34]. However, in the same irreps case, the

allowed pair symmetry has been reported to be only

Egirrep by Ref. [20–23] in D4h[38]. The most sim-

ple basis function of Egsymmetry is the kz(kx+iky)-

wave (i.e., the chiral d-wave). However, this simple

chiral d-wave pairing forms Cooper pairs between

the electrons in diﬀerent layers of an SRO crystal,

which has a nearly two-dimensional electronic struc-

ture; therefore, the formation of such Cooper pairs

has been considered diﬃcult. Nevertheless, when the

orbital degree of freedom is considered, onsite EgSC

pairing states, which can resolve the aforementioned

problems, are possible. Actually, an inter-orbital-

odd spin-triplet s-wave pairing state in the Egirrep

has recently attracted attention [35–37]. This pair

has an energy-gap structure similar to that of a chiral

d-wave pairing but diﬀers in that it has a Bogoliubov

Fermi surface. This state is one of the most promis-

ing candidates because it can explain the µSR ex-

periments under both hydrostatic pressure [39] and

in-plane uniaxial strain [40].

In this paper, we calculate the dispersion of the

surface state and the local density of states (LDOS)

at the (001) surface (hereafter referred to as the top

surface) of the Eginter-orbital SC state by the re-

cursive Green’s function formula [41]. The surface

states appear near zero energy with splits by a much

smaller energy scale than that of the SC gap. The

physical origin of this surface state can be under-

stood by an eﬀective low-energy Hamiltonian char-

acterized by an eﬀective chiral d-wave pair potential

and a pseudo-Zeeman ﬁeld. As a result, the LDOS

has a pronounced zero-energy peak when the order

of the resolution is lower than the splitting energy.

This peak structure is robust against perturbations

at the top surface, such as an orbital Rashba cou-

pling or a chiral p-wave SC pair potential.

arXiv:2210.02136v2 [cond-mat.str-el] 14 Oct 2022

II. ANALYTICAL DESCRIPTION

A. Model Hamiltonian

Let us start with a model Hamiltonian for an inter-

orbital superconducting state of SRO, as originally

proposed in Ref. [36]. We focus on the t2gorbitals

of the Ru ions (i.e., the dyz-, dzx-, and dxy -orbitals),

which dominate the bands near the Fermi level. The

corresponding Bogoliubov–de Gennes (BdG) Hamil-

tonian is given by

H=1

Ψ†

kˇ

H(k)Ψk(1)

with

H(k) = ˆ

HN(k)ˆ

∆(k)

∆†(k)−ˆ

N(−k),(2)

and

Ψ†

k= (C†

k,CT

−k),

C†

k= (c†

k,↑,yz, c†

k,↑,zx, c†

k,↑,xyc†

k,↓,yz, c†

k,↓,zx, c†

k,↓,xy),

(3)

where c†

k,σ,χ creates an electron with momentum k

and spin σin orbital χ. The normal-state Hamilto-

nian is described by

HN(k) = ¯

ξ+¯

λ0+¯

λ3¯

λ1+ i¯

λ2

λ1−i¯

λ2¯

ξ+¯

λ0−¯

λ3,

ξ=



ξyz 0 0

0ξzx 0

0 0 ξxy 

,¯

λ0=



0h10 h20

h10 0h30

h20 h30 0

,

λj=1,2,3=



0−ih4j−ih5j

ih4j0−ih6j

ih5jih6j0

,

(4)

where ¯

ξ+¯

λ0contains the spin-independent hopping

integrals and the chemical potentials and ¯

λj=1,2,3de-

scribe the inter-orbital spin–orbit couplings. The ex-

plicit forms of the matrix elements hij and the cor-

responding band parameters are summarized in Ap-

pendix A. The proposed inter-orbital pair potential

associated with the Egirrep is denoted by

∆ = 0¯

∆

∆ 0 ,¯

∆=∆¯

Lx+ i¯

Ly,

Lx=



000

0 0 i

0−i 0 

,¯

Ly=



0 0 −i

0 0 0

i 0 0 

,

(5)

where ¯

Lxand ¯

Lyare the orbital angular momentum

operators in the orbital space and ∆ (≥0) denotes

the magnitude of the pair potential. We assume

that ∆ is constant with respect to k. The present

pair potential describes spin-triplet superconductiv-

ity, where the d-vector is directed along the c-axis

of SRO. Notably, this inter-orbital superconducting

state is stabilized by the spin–orbit couplings of h53

and h63 [36].

B. Approximate low-energy Hamiltonian

In this subsection, we derive an approximate

Hamiltonian that enables us to grasp the essential

properties of the present model. For this purpose,

although we lose quantitative accuracy, we treat the

inter-orbital hybridizations of ¯

λi=0,1,2,3as the per-

turbation. On the basis of the second-order pertur-

bation theory, we can deform the Hamiltonian in an

approximate band basis as

ˇu†ˇ

H(k)ˇu=



Hα˘

Vαβ ˘

Vαγ

Vβα ˘

Hβ˘

Vβγ

Vγα ˘

Vγβ ˘

Hγ



+O(λ3),(6)

with

Hν=˜εν˜

ψν

ψ†

ν−˜εν,

˜εν=εν˜σ0,˜

ψν=ψν(i˜σ2),

(7)

and

Vνν0=0˜

Dνν0

−˜

D∗

νν00,

Dνν0= (ψνν0+dνν0·˜

σ) (i˜σ2),

dνν0= (d1,νν0, d2,νν0, d3,νν0),

(8)

for ν,ν0=α,β, and γ, where ˜

σ= (˜σ1,˜σ2,˜σ3) and

˜σ0represent the Pauli matrices and the unit matrix

in pseudo-spin space, respectively, and O(λn) repre-

sents the Landau symbol with respect to the n-th or-

der of the matrix elements in ¯

λi. The explicit forms

of the matrix elements and the unitary operator ˇu

are given in Appendix B. We construct the unitary

matrix ˇuto diagonalize the normal-state Hamilto-

nian ˆ

HNwithin the second-order of ¯

λi. In addition,

the diagonal components of εα,εβ, and εγgive the

kinetic energies of three diﬀerent bands, which con-

stitute three separated Fermi surfaces. Thus, the

unitary transformation in Eq. (6) changes the ba-

sis of the Hamiltonian from the original orbital ba-

sis to the approximate band basis. Then, ψν,ν0i˜σ2

(a) (b) (c)

FIG. 1. (Color online) (a) The Fermi surface at kzc= 0, where the arrow shows the kdaaxis, which is the diagonal

line on the kxkyplane. (b) The kzdependence of the Fermi surface on the kdaline shown in ﬁgure (a). Solid lines are

in the approximated band basis, and dotted lines are in the numerical band basis. (c) The numerical Fermi surface

projected into the surface Brillouin zone in the range enclosed by the square shown in ﬁgure (a). Dotted and dashed

lines show the numerical Fermi surface at kzc= 2πand kzc= 0, respectively.

and dν,ν0·˜

σi˜σ2(dν,ν =0) are the spin-singlet

and -triplet pair potentials, respectively. Parame-

ters d1,νν0, d2,νν0,and d3,νν0are the x-, y-, and z-

components of the d-vector, respectively. These pair

potentials have momentum dependence even though

the original pair potential does not. To check the va-

lidity of this approximation, we show the Fermi sur-

faces in the approximate band basis and the numeri-

cal band basis, which are obtained by the numerical

diagonalization of the normal-state Hamiltonian in

Figs. 1(a) and (b). In Fig. 1, the lattice constant

of the conventional unit cell in the in-plane direc-

tions and the z-direction are represented by aand

c, respectively, and cis twice the distance between

nearest-neighbor layers. The approximate Fermi sur-

faces (dotted lines) and the numerical Fermi surfaces

(solid lines) nicely correspond. Figure 1(c) shows the

numerical Fermi surfaces projected onto the surface

Brillouin zone (BZ). The dashed and dotted lines in-

dicate the Fermi surfaces at kzc= 0 and kzc= 2π,

respectively. As we will show later, the property of

the projected Fermi surfaces is related to the emer-

gence of the low-energy surface states at the top sur-

face.

We here proceed with a further approximation to

construct a low-energy eﬀective Hamiltonian for each

band. According to the argument in Refs. [36, 42–

44], the low-energy excitation in the vicinity of the

Fermi surface of the ν-band can be evaluated by

Heﬀ

ν=˘

Hν+X

ν06=ν

V†

ν0ν˘τz˘

Vν0ν

εν−εν0

.(9)

where ˘τz= diag[1,1,−1,−1]. A detailed derivation

of Eq. (9) is presented in Appendix B. Eventually,

we obtain the low-energy eﬀective Hamiltonian for

the ν-band within the second-order perturbation of

λi:

Heﬀ

ν=˜

hν(k)˜

ψν(k)

ψ†

ν(k)−˜

ν(−k),

hν= (εν−γν) ˜σ0+mν·˜

σ,

(10)

with

γν=X

ν06=ν

|ψν0ν|2+|dν0ν|2

εν−εν0

,(11)

mν=X

ν06=ν

2Re[ψν0νd∗

ν0ν]−idν0ν×d∗

ν0ν

ενεν0

,(12)

where γνdescribes the modulation in the kinetic en-

ergy and mνrepresents the pseudo-Zeeman poten-

tial.

We now describe the essential properties of the

present model, which is clariﬁed using the eﬀective

low-energy Hamiltonian ˘

Heﬀ

ν. Remarkably, the ef-

fective pair potential ψνacting on each band can be

decomposed as

ψν=Xν(k)+iYν(k),(13)

where the real functions of Xν(k) and Yν(k) obey

Xν(−kx,ky, kz) = Xν(kx, ky,−kz) = −Xν(k),

Xν(kx,−ky, kz) = Xν(k),(14)

and

Yν(kx,−ky, kz) = Yν(kx, ky,−kz) = −Yν(k),

Yν(−kx, ky, kz) = Yν(k),(15)

respectively. This implies that the eﬀective pair po-

tential ψνhas chiral d-wave pairing symmetry sim-

ilar to the kz(kx+ iky)-wave pair potential. The

pair potential vanishes at kzc= 0 and kzc= 2π.

Thus, in the absence of the pseudo-Zeeman poten-

tial (i.e., mν= 0), ˘

Hνexhibits the line nodes at

kz= 0 and kzc= 2π, where the location of the line

nodes corresponds to the dashed and dotted lines in

Fig. 1(c). In addition, the pair potential has mirror-

odd nature with respect to kz(i.e., ψν(kx, ky, kz) =

−ψν(kx, ky,−kz)), which enables us to expect the

formation of low-energy surface states at the top sur-

face [45–53]. Moreover, it has been already shown

that, for a kz(kx+ iky)-wave superconductor, we

obtain topologically-protected ﬂat-band zero-energy

surface states at the top surface [45, 54, 55]. More

speciﬁcally, the relevant topological invariant pre-

dicts that the ﬂat-band zero-energy surface states

appear at momenta enclosed by the nodal lines in

the surface BZ (see also the detailed discussion in

Appendix D). Thus, on the basis of the analogy be-

tween the kz(kx+ iky)-wave superconductor and the

present superconductor characterized by the eﬀec-

tive chiral d-wave pair potential, we expect that,

if the pseudo-Zeeman potential is absent, ﬂat-band

zero-energy surface states will appear in the momen-

tum range enclosed by the dashed and dotted lines

in Fig. 1(c). Nevertheless, in the present model,

the emergence of the pseudo-Zeeman potential is in-

evitable. The energy eigenvalue of ˘

Heﬀ

νis given by

E±,s =±Ecd +s|mν|,

Ecd =p(εν+γν)2+|ψν|2,(16)

for s=±. The pseudo-Zeeman potential clearly

shifts the bands of ±Ecd, which originally exhibit

the line nodes at momenta satisfying εν+γν= 0

and |ψν|= 0. In particular, the line nodes in

E+,−=Ecd − |mν|and E−,+=−Ecd +|mν|

are inﬂated to the Bogoliubov–Fermi surfaces by

the pseudo-Zeeman ﬁeld [42, 43]. Moreover, we in-

fer that exact ﬂat-band zero-energy surface states

can no longer exist because of the energy splitting

from the pseudo-Zeeman ﬁeld. Even so, the energy

splitting in the surface states would be substantially

smaller than ∆ because the pseudo-Zeeman poten-

tial |mν|is proportional to ∆2.

Summarizing the previous discussion, we can ex-

pect the present model to display nearly zero-energy

surface states at the top surface in the momentum

range enclosed by the dashed and dotted lines in

Fig. 1(c), where the energy splitting of the sur-

face states is due to the pseudo-Zeeman potential

mν∝∆2. In the next section, we will conﬁrm this

statement by examining the surface energy disper-

sion and the surface LDOS numerically.

III. DETAILED PROPERTIES OF

SURFACE STATES

A. Recursive Green’s function techniques

In this section, we consider the open boundary

condition in the z-direction and the periodic bound-

ary condition in the x- and y-directions to calculate

the surface Green’s function for the semi-inﬁnite sys-

tem. In addition, we consider ﬂat surfaces and do not

consider surface reconstructions. Then, the momen-

tum parallel to the surface kk≡(kx, ky) becomes a

good quantum number, and the problem is reduced

to a one-dimensional problem along the z-direction

at each momentum kk. The BdG Hamiltonian ˇ

H(k)

in Eq. (2) includes inter-layer hopping up to the next-

nearest layer. Thus, the Hamiltonian of a system

with nlayers stacked in the z-direction can be writ-

ten as

Hn(kk) =

j=1 X

α,β

C†

j,α,kk{h0(kk)}α,β Cj,β,kk

n−1

j=1 X

α,β

[C†

j,α,kk{t1(kk)}α,β Cj+1,β,kk+ H.c.]

n−2

j=1 X

α,β

[C†

j,α,kk{t2(kk)}α,β Cj+2,β,kk+ H.c.],

(17)

where αand βdenote the internal degrees of free-

dom: spin, orbital, and particle–hole; C†

j,α,kkand

Cj,α,kkare the creation and annihilation operators

at the j-th site in the z-direction, respectively, and

h0(kk) and t1(2)(kk) are the intra-layer element and

the inter-layer hopping between the (next) nearest-

neighbor layers of the Hamiltonian, respectively.

To obtain the surface Green’s function of the top

layer, we use the recursive Green’s function tech-

nique. By formulating the recursion relation us-

ing the M¨obius transformation, we can calculate

the surface Green’s function in a semi-ﬁnite sys-

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SurfaceStateofInter-orbitalPairingStateinSr2RuO4SuperconductorSatoshiAndo,SatoshiIkegaya,ShunTamura,YukioTanaka,KeijiYadaDepartmentofAppliedPhysics,NagoyaUniversity,Nagoya464-8603,Japan(Dated:October17,2022)Westudythe(001)surfacestateofarecentlyproposedEgsymmetryinter-orbital-oddspin-triplets-wavesu...

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Surface State of Inter-orbital Pairing State in Sr2RuO 4Superconductor Satoshi Ando Satoshi Ikegaya Shun Tamura Yukio Tanaka Keiji Yada Department of Applied Physics Nagoya University Nagoya 464-8603 Japan

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