Second-order Topological Superconductors with Mixed Pairing

Xiaoyu ZhuSchool of Science, MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of Condensed Matter,

Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China(Dated: June 18, 2019)

We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driveninto the second-order topological superconducting phase when a mixed-pairing state is introduced.The superconducting order we consider involves only even-parity components and meanwhile breakstime-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor wouldhost one single Majorana zero mode in the second-order nontrivial phase. Starting from edgephysics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces andnodal points of the superconducting order parameter. In the end, we propose two setups that maybring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on theexperimental feasibility of the two platforms.

Topological superconductors (TSCs) distinguish them-selves from trivial ones in the robust midgap states—Majorana zero modes (MZMs)—that could form eitherat local defects or boundaries [1–9]. Among the variousproposals for TSCs, semiconducting systems with Rashbaspin-orbit coupling (RSOC) [10–13] as well as topologi-cal insulating systems [14] have attracted the most atten-tion. In both platforms, signatures of MZMs have beenobserved when conventional s-wave pairing is introducedthrough proximity effect [15–25].

In these conventional, also termed as first-order, TSCs,topologically nontrivial bulk in d dimensions is usuallyaccompanied by MZMs confined at (d − 1)-dimensionalboundaries, the so-called bulk-boundary correspondence.Very recently, this correspondence was extended in topo-logical phases of nth order [26–45], where topologicallyprotected gapless modes emerge at (d − n)-dimensionalboundaries. In Refs.[46–49], the authors demonstratethat a topological insulator could be transformed intoa second-order TSC when unconventional pairing withthe s±- or dx2−y2-wave form is introduced. Looking backat the history of first-order TSCs, one may then ask ifit is possible for a Rashba semiconductor (RS), whichis itself a trivial system as opposed to topological insu-lators, to accommodate such a higher-order nontrivialphase as well. In this work, we will show that it is pos-sible, provided a mixed-pairing state that exhibits bothextended s-wave and dx2−y2-wave symmetries could beinduced therein.

Admixture of the two aforementioned pairing stateswas envisioned shortly after the discovery of iron-basedsuperconductors (FeSCs) [50–53]. Since then tremendousefforts have been made to identify this mixed-pairing or-der [54, 55]. In this Letter, we shall consider a generalmixed state that could reduce to three intensively stud-ied mixed pairings in FeSCs, that is, s+d [56], s+ is [57]and s+ id [50, 52]. Our main finding is that, such a pair-ing state alone could possibly drive a two-dimensional RSinto the second-order topological superconducting phase.

mixed-pairing SCRS

MZMA

B C

D(a)

-1 0 1 2 3-1

0

1

I II

III

(c)

(b)

Cuprate SCFeSC

MZM

RS

FIG. 1. (a). Heterostructure of a RS and SC with mixed pair-ing (top view). Majorana zero modes (denoted by red solidellipses) emerge at the four corners in the second-order topo-logical superconducting phase. Two sets of coordinate sys-tems connected by C4 rotation are shown, in both real spaceand reciprocal space. (b). Hybrid Josephson junction withFeSC (s± pairing), cuprate SC (dx2−y2 pairing), and a singleRS layer sandwiched between them. The junction interface isparallel to the ab-plane of the two SCs. (c). Phase diagramin absence of ∆sd, with ∆0 = ∆1 = 1. Phase I: first-orderTSC; Phase II: nodal SC; Phase III: fully gapped trivial SC.For s + id pairing, Phase I and II would be driven into thesecond-order phase. All parameters in this and the followingfigures are in the unit of t.

Of the three specific forms aforementioned, however, onlys+ id pairing could make it. An accurate criterion is fur-ther established for the second-order phase to emerge,which is closely related to the relative position betweennodal points of the pairing order parameter and the twonondegenerate Fermi surfaces split by RSOC.

We consider a RS in two dimensions with mixed pair-ing of extended s-wave and dx2−y2 -wave form, and the

arX

iv:1

812.

0889

6v2

[co

nd-m

at.m

es-h

all]

15

Jun

2019

2

corresponding Hamiltonian is given by

H =1

2

∑k

Ψ†(k)H(k)Ψ(k),

H(k) = h(k)τ3 + ∆s(k)τ1 + ∆sd(k)τ2, (1)

in the Nambu spinor basis Ψ(k) = {ck↑, ck↓, c†k↓,−c†k↑}T .

In Eq. (1), h(k) = 2A(sin kxσ2 − sin kyσ1)− 2t(cos kx +cos ky) − µ, with t, A and µ being hopping amplitude,RSOC strength and chemical potential respectively, andPauli matrices σ1,2,3, τ1,2,3 act in spin and Nambu spaceseparately. The superconducting term ∆s(k) = ∆0 +2∆1(cos kx+cos ky) in Eq. (1), denoting extended s-wavepairing, and ∆sd(k) = −2∆2(cos kx + δ cos ky + η), de-scribing s+d pairing that in addition exhibits a π/2-phaseshift relative to ∆s. This time-reversal-symmetry(TRS)-broken pairing reduces to s + d when ∆s = 0, to s + idwhen δ = −1, η = 0, and to s + is when δ = 1. Theenergy spectrum of Hamiltonian Eq. (1) has a simpleform,

E(k) = ±√ε2±(k) + ∆2

s(k) + ∆2sd(k), (2)

where ε±(k) = ±2A√

sin2 kx + sin2 ky − 2t(cos kx +

cos ky)− µ, being the kinetic energy.

In the absence of ∆sd term, the model is well knownto support first-order topological superconducting phasethat features TRS-protected helical Majorana modes onthe edges, as well as nodal superconducting phase withpoint nodes [58] (see Fig. 1(c)), provided

|µ− 4tα∆| < 2√

2|A|√

1− α2∆, |α∆| < 1 (3)

with α∆ = ∆0/(4∆1). Equation (3) can be fulfilledwhen the system exhibits s± pairing symmetries. Turn-ing on ∆sd is supposed to break TRS and gap out thehelical modes. Instead of driving the system into trivialphases, we will demonstrate that this TRS-broken termmay give birth to second-order topological superconduct-ing phases, featuring MZMs bound at corners. To un-derstand the origin of second-order phases, we may startfrom gapless edge states in the absence of ∆sd and thenconsider effects of this mass term on the gapless modes.

As is known, second-order phases appear when gaplessstates on intersecting edges acquire mass gaps of oppo-site signs. To investigate the edge physics, we considera cylinder geometry, where the periodic boundary con-dition is only assumed along the y direction [see Fig.1(a)]. Accordingly, the Hamiltonian in this geometrywould take the form H = 1

2

∑ky

Ψ†(ky)H1D(ky)Ψ(ky),

when written in the new basis Ψ(ky) = ⊕jψj(ky), where

ψj(ky) = {cj,ky↑, cj,ky↓, c†j,−ky↓,−c

†j,−ky↑}

T , cj,ky↑(↓) =1√Nx

∑kxck↑(↓)e

ikxj , j stands for lattice site and Nx is

the total number of sites. The components of H1D(ky)

are given by the following 4× 4 block matrices,

[H1D(ky)]j,j = M = MαβΓαβ , (4)

[H1D(ky)]j,j+1 = ([H1D(ky)]j+1,j)† = T = TαβΓαβ .

In Eq. (4) Γαβ = τα ⊗ σβ with α, β = 0, 1, 2, 3, andthe two tensors M and T have the following entries:M30 = −µ − 2t cos ky, M31 = −2A sin ky, M10 =∆0 + 2∆1 cos ky, M20 = −2∆2(η + δ cos ky), T 30 = −t,T 32 = −iA, T 10 = ∆1 and T 20 = −∆2. The energyspectrum and corresponding wave functions in this ge-ometry could be determined from the eigenvalue equationH1D(ky)φ = E(ky)φ, which leads to

Mφj + T†φj−1 + Tφj+1 = E(ky)φj , for any j, (5)

with φj being a four-component vector that representsthe wave function at site j.

In the first-order phase when ∆sd = 0, we have E(ky =π) = 0 if α∆ > 0 and E(ky = 0) = 0 otherwise [58].In both cases MZMs are doubly degenerate on edge ABas well as CD defined in Fig. 1(a). Without loss ofgenerality, hereafter we will assume α∆ > 0. In the nodalphase, zero modes in the spectrum E(ky) would appearat the projections of bulk nodes on the edge Brillouinzone (BZ), as is shown in Figs. 2(a) and 2(c). There areeight nodes in total, which relate to one another throughfourfold rotation C4, mirror reflectionsMx andMy withmirror planes at kx = 0 and ky = 0. In the absenceof ∆sd, Hamiltonian Eq. (1) is invariant under these

BZ

0 π-π

0

Site

0 1Scaled wave function amplitude-1

0

1

-5 -2 1 4 7

(c)

(d)

(b)

(a)

FIG. 2. Edge states in the nodal phase. (a) Bulk nodes inBZ are denoted by stars, and those in the same color have thesame topological charge. (b) Winding number for topologi-cally distinct nodes and its evolution with chemical potential.Clearly, only in the nodal phase would the winding numbertake nonzero values. (c) Energy spectrum in the cylinder ge-ometry. Red lines denote the four energy levels closest tozero and each of the lines is doubly degenerate. (d) Varia-tions of the (scaled) wave function amplitude |φj | with latticesite j and wave vector ky for the energy levels denoted by redlines in (c). The parameters chosen are, Nx = 200, A = 2,∆0 = ∆1 = 1, ∆2 = 0. In (c) and (d), µ = 5.

3

operations, i.e.,

U−1C4 H(kx, ky)UC4 = H(−ky, kx),

U−1MxH(kx, ky)UMx

= H(−kx, ky),

U−1MyH(kx, ky)UMy

= H(kx,−ky), (6)

where UC4 = eiπσ3/4, UMx= σ1, and UMy

= σ2. Becauseof these crystalline symmetries, we may denote the eightbulk nodes by ±(k+,±k−) and ±(k−,±k+), with

cos k± = −α∆ ±√

1− α2∆ − (µ− 4tα∆)2/(8A2). (7)

In contrast to the first-order phase, zero modes at ±k± inthe nodal phase are not localized. However, in the edgeBZ where ky ∈ (−k−,−k+)∪ (k+, k−) (0 < k+ < k− < πis assumed), we find that two localized states with op-posite excitation energy ±E(ky) would exist on eachedge, as is evidenced in Figs. 2(c) and 2(d). It seemsthat these edge states are not topologically protected,since each gapless point (±k±) is the projection of twobulk nodes carrying opposite topological charges [see Fig.2(a)] which are supposed to cancel each other out. Thecharge for each bulk node is defined by the winding num-ber w along a contour l surrounding this node [59], as isshown in Fig. 2(a) (see Supplemental Material for de-tails). Possibly, these localized edge states are the rem-nants of those in the first-order phase. In our specificmodel defined in Eq. (1), they are robust provided thesystem is in the nodal phase. Hence we may describe thelow-energy physics of each edge with a gapless Hamilto-nian that is defined only at ky ∈ (−k−,−k+) ∪ (k+, k−).

So we have established that MZMs emerge in boththe first-order and the nodal phase when ∆sd = 0.Edge states in these two phases could be well describedby a one-dimensional massless Hamiltonian, with gap-less points at kcy = π in the first-order phase, and atkcy = ±k± in the nodal phase. Note that Hamilto-nian Eq. (5) preserves chiral symmetry Γ20 in the ab-sence of ∆sd, which guarantees that, for any state φwith finite energy E(ky) there would be a state Γ20φ(shorthand for ⊕jΓ20φj) with opposite energy −E(ky).Hence we can define the MZM basis for each edge as{φ(kcy),Γ20φ(kcy)}T . Instead of going into the details ofMZMs, we will attempt to construct an effective edgeHamiltonian with a unified form.

First, multiplying Eq. (5) with φ†jΓ10 on both sides,summing over j and then adding to it the Hermitian con-jugating counterpart, we are then left with

M10 =∑j

E(ky)φ†jΓ10φj − T 10φ†j(φj−1 + φj+1), (8)

where the normalization condition φ†φ = 1 is used. Onecould also multiply Eq. (5) with φ†jΓ20Γ10, and followthe same procedure as above, which would lead to

T 10∑j

φ†jΓ20(φj−1 + φj+1) = 0, (9)

due to orthogonality condition φ†Γ20φ = 0. At the gap-less point kcy, Eq. (8) reduces to

M10 = −T 10∑j

φ†j(φj−1 + φj+1). (10)

After projecting Hamiltonian Eq. (4) onto the MZM ba-sis and utilizing the two equalities in Eqs. (9) and (10),one arrives at the effective low-energy Hamiltonian foredge AB or CD, given by

HEdge(ky) = v2(ky)s2 + v3(ky)s3 +msd(ky)s1, (11)

where

v2(ky) =∑j,{αβ}

[Mαβ(ky)−Mαβ(kcy)]φ†jΓ20Γαβφj ,

v3(ky) =∑j,{αβ}

[Mαβ(ky)−Mαβ(kcy)]φ†jΓαβφj ,

msd(ky) = −2∆2(δ cos ky + η − cos kcy − 2α∆), (12)

with indices {αβ} taking {30, 31, 10} and Pauli matricess1,2,3 acting in the MZM basis. Wave functions of MZMs— φj in Eq. (12) — could be obtained by solving Eq.(5) in principle, although we don’t have to, given that itis the mass gap msd that we care foremost, and that itclearly doesn’t depend on the specific form of φj . Withthe edge Hamiltonian Eq. (11) being given, the conditionwhen second-order phases emerge can be determined bycomparing signs of mass gaps on intersecting edges, whichwe shall detail in the following.

Let us consider rotating the basis in Eq. (1) toΨ′(k′) = UC4Ψ(C4k′), where k′ stands for coordinatesin the O − k′xk′y system defined in Fig. 1(a) and relatesto k through C4 rotation C4k′ = k, namely, (−k′y, k′x) =(kx, ky). Rewriting Hamiltonian Eq. (1) in this new ba-sis, we would have

H =1

2

∑k′

Ψ′†(k′)H′(k′)Ψ′(k′), (13)

H′(k′) = UC4H(C4k′)U−1C4 = h(k′) + ∆s(k

′) + ∆′sd(k′),

where ∆′sd(k′) = −2∆2(cos k′y +δ cos k′x+η) and the last

equality in Eq. (13) is due to C4 symmetry of h and ∆s

detailed in Eq. (6). Comparing the two Hamiltoniansin Eqs.(1) and (13), one may immediately conclude thatthe edge Hamiltonian along edge AD or BC could beobtained from Eq. (11) simply by replacing ky with k′y,followed by modification of the mass term, which yields

H′Edge(k′y) = v2(k′y)s2 + v3(k′y)s3 +m′sd(k′y)s1, (14)

with

m′sd(k′y) = −2∆2(cos k′y + η − δ cos kcy − 2δα∆), (15)

and the definitions of v2 and v3 are given in Eq. (12). It isobvious that gapless points in the two edge Hamiltonian,

4

0

π/2

π

0

π/2

π

0 π/2 π 0 π/2 π

(a) (b)

(d)(c)

FIG. 3. Determination of the second-order phase from thebulk spectrum. (a)-(d) Fermi surfaces ε± = 0, nodal lines of∆s, ∆sd and ∆′sd in the first quadrant of BZ are plotted fordifferent chemical potential (µ) and s+ d pairing form (δ, η).Signs of the pair [∆sd(k),∆′sd(k)] are indicated in correspond-ing areas. The system resides in second-order phases when thesigns of ∆sd and ∆′sd at kc (marked by stars in magenta) areopposite, or equivalently, when the nodal point kn (markedby magenta circle) of the pairing term lies between the twoFermi surfaces. Distributions of MZMs for an 80× 80 latticeare shown in the insets, as well as several low-lying energylevels. Clearly, MZMs (red points in the insets) are fourfolddegenerate and separated from other energy levels with a fi-nite gap. In all the figures, A = ∆2 = 0.5, ∆0 = ∆1 = 1.

HEdge(ky) and H′Edge(k′y), both reside at kcy. The second-order phase therefore emerges when

msd(kcy)m′sd(k

cy) < 0. (16)

Additionally, we require Eq. (3) to be fulfilled, whichguarantees that the system falls into the first-order ornodal phase when ∆sd is switched off.

Further investigations on Eqs. (12) and (15) revealthat, the mass terms msd(k

cy) and m′sd(k

cy) are nothing

but values of ∆sd(k) and ∆′sd(k′) at point kc = (kcx, k

cy)

that satisfies ∆s(kc) = 0, with kcy being the gapless point

in the edge BZ. Thus we may relate the criterion obtainedfrom the edge Hamiltonian with the bulk spectrum in Eq.(2). As illustrated in Fig. 3, Eq. (16) actually requires∆sd and ∆′sd to take opposite signs at kc marked by stars,that is,

∆sd(kc)∆′sd(k

c) < 0. (17)

Substituting the expression of kcy into Eq. (16), we arrive

at the conditions for second-order phases,

|η − f1| < |f3|, (18)

|µ− 4tα∆| < |2√

2A

1− δ|√f2

2 − (η − f1)2, (19)

with f1 = (1 + δ)α∆, f2 = (1 − δ)√

1− α2∆ and

f3 = (1 − δ)(1 − α∆). Equation (18) determines whichkind of pairing form could possibly induce the second-order phase, while Eq. (19) establishes the relation ofFermi surfaces with the pairing potential in this nontriv-ial phase. Indeed, we observe that the nodal point kn

(∆s(kn) = ∆sd(k

n) = 0) of the superconducting orderparameter, marked by a magenta circle in Fig. 3, alwayslies between the two Fermi surfaces in the second-orderphase. This is verified by the fact that Eq. (19) couldalso be obtained by requiring

ε+(kn)ε−(kn) < 0, (20)

where ε± are the same as those in Eq. (2) and take zeroseparately on the two Fermi surfaces. In addition, we alsonote that Eq. (18) actually guarantees the existence ofnodal point kn. Therefore, one may determine when thesystem resides in the second-order phase, either from Eqs.(18) and (19), or from Eq. (20), as illustrated in Fig. 3.Following these criteria, one may immediately concludethat s + id pairing favors the second-order phase whileneither s+ d nor s+ is pairing do.

The mixed-pairing state we consider above has beenextensively studied in iron pnictides, particularly 122compounds [60–66] like Ba1−xKxFe2As2. In these ma-terials, the pairing symmetry is expected to change froma nodeless s± form around optimal doping (x ∼ 0.4)[67] to a form with nodal gaps in the heavily hole-dopedregion, for instance, KFe2As2 (x = 1). In a narrow dop-ing region between the two cases, two different pairingstates may coexist with an additional π/2-phase shift atlower temperature when TRS is spontaneously break-ing [52, 65, 68]. The main debate, however, centersaround the heavily hole-doped region, where multiple ex-periments suggest contradicting pairing, either nodal s-[69, 70] or d wave [60, 63, 71, 72]. Accordingly, the in-termediate state would exhibit either s + is or s + idsymmetry, as was reported in muon spin rotation experi-ment at doping level around x = 0.73 [73]. No consensushas been achieved as to which specific form it would take,although several proposals have been put forward to dis-criminate the two mixed pairings [74–76]. In this regard,our study suggests an alternative approach to tackle thisissue, given that s+ id could drive a RS into the second-order phase with MZMs sitting at corners whereas s+ ispairing couldn’t. Once s+ id pairing has been confirmed,it would be straightforward to fabricate the heterostruc-ture as depicted in Fig. 1(a) and to investigate MZMsbeing expected therein.

Meanwhile, we may also consider a hybrid Josephsonjunction, as schematically shown in Fig. 1(b). The FeSC

5

on top with s± pairing and the cuprate SC at the bottomwith dx2−y2 pairing may introduce a mixed state of theform s+ eiθd in the RS layer sandwiched between them.In the Supplemental Material we demonstrate that thiskind of pairing falls into the general form studied above,and interestingly it always favors second-order phases ex-cept when the phase difference θ of the two SCs takes0 or π, which corresponds to s ± d pairing. To guar-antee that the phase difference never takes 0 or π, onemay insert this hybrid system into a single-junction rfSQUID [77] or a two-junction dc SQUID [78], where θmay be tuned through magnetic flux threaded into theinterferometer. Actually, it has been suggested that sucha hybrid system could naturally realize a junction withθ = π/2 and thus s + id pairing order would develop atthe interface [79]. Hybrid Josephson junctions containingconventional s-wave SCs and FeSCs [80, 81] or cuprateSCs [82, 83] have been successfully fabricated and wellstudied. We can therefore expect the hybrid junctionwith an FeSC, RS, and cuprate SC to be a promisingplatform for second-order TSCs in the near future.

The author acknowledges J. Liu for many illuminatingdiscussions. This work was supported by National Nat-ural Science Foundation of China (NSFC) under GrantNo. 11704305 and No. 11847236.

[1] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).[2] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).[3] F. Wilczek, Nat Phys 5, 614 (2009).[4] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057

(2011).[5] J. Alicea, Reports on Progress in Physics 75, 076501

(2012).[6] C. W. J. Beenakker, Annual Review of Condensed Matter

Physics 4, 113 (2013).[7] T. D. Stanescu and S. Tewari, Journal of Physics: Con-

densed Matter 25, 233201 (2013).[8] S. R. Elliott and M. Franz, Rev. Mod. Phys. 87, 137

(2015).[9] R. Aguado, La Rivista del Nuovo Cimento 40, 523

(2017).[10] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,

Phys. Rev. Lett. 104, 040502 (2010).[11] J. Alicea, Phys. Rev. B 81, 125318 (2010).[12] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.

Lett. 105, 077001 (2010).[13] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.

105, 177002 (2010).[14] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407

(2008).[15] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.

A. M. Bakkers, and L. P. Kouwenhoven, Science 336,1003 (2012).

[16] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, andH. Shtrikman, Nat Phys 8, 887 (2012).

[17] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson,P. Caroff, and H. Q. Xu, Nano Letters 12, 6414 (2012).

[18] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat Phys8, 795 (2012).

[19] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni,K. Jung, and X. Li, Phys. Rev. Lett. 110, 126406 (2013).

[20] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon,M. Leijnse, K. Flensberg, J. Nygard, P. Krogstrup, andC. M. Marcus, Science 354, 1557 (2016).

[21] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon,J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yaz-dani, Science 346, 602 (2014).

[22] S. Hart, H. Ren, T. Wagner, P. Leubner, M. Muhlbauer,C. Brune, H. Buhmann, L. W. Molenkamp, and A. Ya-coby, Nat Phys 10, 638 (2014).

[23] J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang,C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu,Q.-H. Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia,Phys. Rev. Lett. 114, 017001 (2015).

[24] H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang,H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-D. Guan, Y.-Y. Li,C. Liu, D. Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang,and J.-F. Jia, Phys. Rev. Lett. 116, 257003 (2016).

[25] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che,G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Mu-rata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C.Zhang, K. Liu, J. Xia, and K. L. Wang, Science 357,294 (2017).

[26] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,Science 357, 61 (2017).

[27] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,Phys. Rev. B 96, 245115 (2017).

[28] F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook,A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock,S. Jeon, I. Drozdov, H. Bouchiat, S. Gueron, A. Yazdani,B. A. Bernevig, and T. Neupert, Nature Physics 14, 918(2018).

[29] Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. 119,246402 (2017).

[30] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, andP. W. Brouwer, Phys. Rev. Lett. 119, 246401 (2017).

[31] M. Ezawa, Phys. Rev. Lett. 120, 026801 (2018).[32] E. Khalaf, Phys. Rev. B 97, 205136 (2018).[33] M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer,

Phys. Rev. B 97, 205135 (2018).[34] X. Zhu, Phys. Rev. B 97, 205134 (2018).[35] Y. You, T. Devakul, F. J. Burnell, and T. Neupert, Phys.

Rev. B 98, 235102 (2018).[36] Y. Volpez, D. Loss, and J. Klinovaja, arXiv e-prints

, arXiv:1811.01827 (2018), arXiv:1811.01827 [cond-mat.mes-hall].

[37] M. Serra-Garcia, V. Peri, R. Susstrunk, O. R. Bilal,T. Larsen, L. G. Villanueva, and S. D. Huber, Nature555, 342 (2018).

[38] C. W. Peterson, W. A. Benalcazar, T. L. Hughes, andG. Bahl, Nature 555, 346 (2018).

[39] S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W.Molenkamp, T. Kiessling, F. Schindler, C. H. Lee,M. Greiter, T. Neupert, and R. Thomale, Nature Physics14, 925 (2018).

[40] J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P.Chen, T. L. Hughes, and M. C. Rechtsman, Nature Pho-tonics 12, 408 (2018).

[41] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang,S. S. P. Parkin, B. A. Bernevig, and T. Neupert, ScienceAdvances 4, eaat0346 (2018).

6

[42] R.-J. Slager, L. Rademaker, J. Zaanen, and L. Balents,Phys. Rev. B 92, 085126 (2015).

[43] D. Calugaru, V. Juricic, and B. Roy, Phys. Rev. B 99,041301 (2019).

[44] Y. Wang, M. Lin, and T. L. Hughes, Phys. Rev. B 98,165144 (2018).

[45] Z. Wu, Z. Yan, and W. Huang, Phys. Rev. B 99, 020508(2019).

[46] Z. Yan, F. Song, and Z. Wang, Phys. Rev. Lett. 121,096803 (2018).

[47] Q. Wang, C.-C. Liu, Y.-M. Lu, and F. Zhang, Phys.Rev. Lett. 121, 186801 (2018).

[48] T. Liu, J. J. He, and F. Nori, Phys. Rev. B 98, 245413(2018).

[49] R.-X. Zhang, W. S. Cole, and S. Das Sarma, arXive-prints , arXiv:1812.10493 (2018), arXiv:1812.10493[cond-mat.supr-con].

[50] W.-C. Lee, S.-C. Zhang, and C. Wu, Phys. Rev. Lett.102, 217002 (2009).

[51] C. Platt, R. Thomale, C. Honerkamp, S.-C. Zhang, andW. Hanke, Phys. Rev. B 85, 180502 (2012).

[52] M. Khodas and A. V. Chubukov, Phys. Rev. Lett. 108,247003 (2012).

[53] R. M. Fernandes and A. J. Millis, Phys. Rev. Lett. 110,117004 (2013).

[54] A. Chubukov and P. J. Hirschfeld, Physics Today 68, 46(2015).

[55] R. M. Fernandes and A. V. Chubukov, Reports onProgress in Physics 80, 014503 (2017).

[56] G. Livanas, A. Aperis, P. Kotetes, and G. Varelogiannis,Phys. Rev. B 91, 104502 (2015).

[57] S. Maiti and A. V. Chubukov, Phys. Rev. B 87, 144511(2013).

[58] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett.111, 056402 (2013).

[59] A. P. Schnyder and S. Ryu, Phys. Rev. B 84, 060504(2011).

[60] F. F. Tafti, A. Juneau-Fecteau, M.-E. Delage,S. Renede Cotret, J.-P. Reid, A. F. Wang, X.-G. Luo,X. H. Chen, N. Doiron-Leyraud, and L. Taillefer, Na-ture Physics 9, 349 (2013).

[61] F. Kretzschmar, B. Muschler, T. Bohm, A. Baum,R. Hackl, H.-H. Wen, V. Tsurkan, J. Deisenhofer, andA. Loidl, Phys. Rev. Lett. 110, 187002 (2013).

[62] T. Bohm, A. F. Kemper, B. Moritz, F. Kretzschmar,B. Muschler, H.-M. Eiter, R. Hackl, T. P. Devereaux,D. J. Scalapino, and H.-H. Wen, Phys. Rev. X 4, 041046(2014).

[63] F. F. Tafti, A. Ouellet, A. Juneau-Fecteau, S. Faucher,M. Lapointe-Major, N. Doiron-Leyraud, A. F. Wang, X.-G. Luo, X. H. Chen, and L. Taillefer, Phys. Rev. B 91,054511 (2015).

[64] Z. Guguchia, A. Amato, J. Kang, H. Luetkens, P. K.Biswas, G. Prando, F. von Rohr, Z. Bukowski, A. Shen-gelaya, H. Keller, E. Morenzoni, R. M. Fernandes, andR. Khasanov, Nature Communications 6, 8863 (2015).

[65] Z. Guguchia, R. Khasanov, Z. Bukowski, F. von Rohr,M. Medarde, P. K. Biswas, H. Luetkens, A. Amato, andE. Morenzoni, Phys. Rev. B 93, 094513 (2016).

[66] J. Li, P. J. Pereira, J. Yuan, Y.-Y. Lv, M.-P. Jiang,D. Lu, Z.-Q. Lin, Y.-J. Liu, J.-F. Wang, L. Li, X. Ke,G. Van Tendeloo, M.-Y. Li, H.-L. Feng, T. Hatano, H.-B.Wang, P.-H. Wu, K. Yamaura, E. Takayama-Muromachi,

J. Vanacken, L. F. Chibotaru, and V. V. Moshchalkov,Nature Communications 8, 1880 (2017).

[67] H. Hosono and K. Kuroki, Physica C: Superconductivityand its Applications 514, 399 (2015), superconductingMaterials: Conventional, Unconventional and Undeter-mined.

[68] C. Platt, G. Li, M. Fink, W. Hanke, and R. Thomale,physica status solidi (b) 254, 1600350 (2017).

[69] K. Okazaki, Y. Ota, Y. Kotani, W. Malaeb, Y. Ishida,T. Shimojima, T. Kiss, S. Watanabe, C.-T. Chen, K. Ki-hou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa,Y. Kohori, K. Hashimoto, T. Shibauchi, Y. Matsuda,H. Ikeda, H. Miyahara, R. Arita, A. Chainani, andS. Shin, Science 337, 1314 (2012).

[70] K. Cho, M. Konczykowski, S. Teknowijoyo, M. A.Tanatar, Y. Liu, T. A. Lograsso, W. E. Straszheim,V. Mishra, S. Maiti, P. J. Hirschfeld, and R. Prozorov,Science Advances 2 (2016), 10.1126/sciadv.1600807.

[71] M. Abdel-Hafiez, V. Grinenko, S. Aswartham, I. Mo-rozov, M. Roslova, O. Vakaliuk, S. Johnston, D. V.Efremov, J. van den Brink, H. Rosner, M. Kumar,C. Hess, S. Wurmehl, A. U. B. Wolter, B. Buchner, E. L.Green, J. Wosnitza, P. Vogt, A. Reifenberger, C. Enss,M. Hempel, R. Klingeler, and S.-L. Drechsler, Phys. Rev.B 87, 180507 (2013).

[72] V. Grinenko, D. V. Efremov, S.-L. Drechsler,S. Aswartham, D. Gruner, M. Roslova, I. Morozov,K. Nenkov, S. Wurmehl, A. U. B. Wolter, B. Holzapfel,and B. Buchner, Phys. Rev. B 89, 060504 (2014).

[73] V. Grinenko, P. Materne, R. Sarkar, H. Luetkens, K. Ki-hou, C. H. Lee, S. Akhmadaliev, D. V. Efremov, S.-L.Drechsler, and H.-H. Klauss, Phys. Rev. B 95, 214511(2017).

[74] S. Maiti, M. Sigrist, and A. Chubukov, Phys. Rev. B 91,161102 (2015).

[75] S.-Z. Lin, S. Maiti, and A. Chubukov, Phys. Rev. B 94,064519 (2016).

[76] J. Garaud, M. Silaev, and E. Babaev, Phys. Rev. Lett.116, 097002 (2016).

[77] P. V. Komissinski, E. Ilichev, G. A. Ovsyannikov,S. A. Kovtonyuk, M. Grajcar, R. Hlubina, Z. Ivanov,Y. Tanaka, N. Yoshida, and S. Kashiwaya, EurophysicsLetters (EPL) 57, 585 (2002).

[78] T. Katase, Y. Ishimaru, A. Tsukamoto, H. Hiramatsu,T. Kamiya, K. Tanabe, and H. Hosono, SuperconductorScience and Technology 23, 082001 (2010).

[79] Z. Yang, S. Qin, Q. Zhang, C. Fang, and J. Hu, Phys.Rev. B 98, 104515 (2018).

[80] X. Zhang, Y. S. Oh, Y. Liu, L. Yan, K. H. Kim, R. L.Greene, and I. Takeuchi, Phys. Rev. Lett. 102, 147002(2009).

[81] A. A. Kalenyuk, A. Pagliero, E. A. Borodianskyi, A. A.Kordyuk, and V. M. Krasnov, Phys. Rev. Lett. 120,067001 (2018).

[82] R. Kleiner, A. S. Katz, A. G. Sun, R. Summer, D. A.Gajewski, S. H. Han, S. I. Woods, E. Dantsker, B. Chen,K. Char, M. B. Maple, R. C. Dynes, and J. Clarke, Phys.Rev. Lett. 76, 2161 (1996).

[83] P. Komissinskiy, G. A. Ovsyannikov, I. V. Borisenko,Y. V. Kislinskii, K. Y. Constantinian, A. V. Zaitsev, andD. Winkler, Phys. Rev. Lett. 99, 017004 (2007).

1

Supplemental Material for “Second-ordertopological superconductors with mixed pairing”

A. Winding number in the nodal phase

In the nodal superconducting phase of our two-dimensional system, there are eight point nodes sittingin the Brillouin zone. For each node one can define thetopological charge as the winding number of a closed loopthat encircles only this particular gapless point. Addi-tionally, we require the loop not to cross any other node,so that the winding number can be well defined over theloop. In the following we shall illustrate how to calcu-late the winding number and corresponding topologicalcharge of the point node for our given system (one mayrefer to Ref. [59] and the supplemental material thereinfor a detailed and general analysis).

Note that in the absence of ∆sd term, our model Hamil-tonian reduces to

H(k) = h(k)τ3 + ∆s(k)τ1. (1)

Owing to the chiral symmetry S = τ2, this Hamiltoniancan be brought into block off-diagonal form, which reads

H(k) = U−1H(k)U =

(0 D(k)

D†(k) 0

), (2)

with U = eiπ3 ·

1√3

(τ1+τ2+τ3), and D(k) = h(k)− i∆s(k)12,

being a 2× 2 complex matrix. Since the loop we choosedoesn’t cross any node, energy spectrum is thus gappedeverywhere on it. In this case, it would be very conve-nient to work with a spectrally flatten Hamiltonian whilecalculating the winding number over a given loop. Thespectrally flatten Hamiltonian takes the similar form asin Eq. (2), being

Q(k) =

(0 q(k)

q†(k) 0

), q(k) = U(k)V †(k). (3)

U(k) and V †(k) in Eq. (3) are 2×2 unitary matrices andrelate to D(k) through singular-value decomposition,

D(k) = U(k)D(k)V †(k), (4)

with D(k) being a diagonal matrix where all the entriesare real and positive (note that det[D(k)] 6= 0 for all kon the loop). Clearly, q(k) is also a 2× 2 unitary matrixand is well defined on the two-dimensional BZ except atthe eight point nodes.

In general, U and V may be expressed using eigenstatesof D†D or DD†. Denote the eigenstates of D†D by ϕn,with corresponding eigenvalues being ξn, and we have

D†Dϕn = ξnϕn, n = 1, 2. (5)

After subsitituting Eq. (4) into Eq. (5), we arrive at

D2V †ϕn = ξnV†ϕn. (6)

Apparently, ξn are the eigenvalues of diagonal matrix D2

and V †ϕn the eigenstates. As a result, D2 can be writtenas

D2 =

(ξ1 00 ξ2

), (7)

and we may choose

V †ϕ1 =

(10

), V †ϕ2 =

(01

), (8)

that is,

V †(ϕ1 ϕ2

)= 12. (9)

Due to the orthogonality and normalization of ϕn, wethus have

V =(ϕ1 ϕ2

), V † =

(ϕ†1ϕ†2

). (10)

The unitary matrix U can be obtained using the relationin Eq. (4), and takes the form

U = DV D−1 =(ξ−11 Dϕ1 ξ−1

2 Dϕ2

). (11)

The q-matrix can then be readily obtained, which reads

q(k) = UV † = ξ−11 Dϕ1ϕ

†1 + ξ−1

2 Dϕ2ϕ†2. (12)

Alternatively, one may also express q-matrix with eigen-states ϕn of DD†, where

DD†ϕn = ξnϕn. (13)

Following the same procedure as above, we would have

q(k) = UV † = ξ−11 ϕ1ϕ

†1D + ξ−1

2 ϕ2ϕ†2D. (14)

Note that there is a freedom in multiplying ϕn(ϕn) by aphase factor eiθn when ξ1 6= ξ2, which obviously doesn’talter the form of q(k). If the spectrum is degenerate, i.e.,ξ1 = ξ2 = ξ, Eq. (12) would simply reduce to

q(k) = ξ−1D(k), (15)

from which it follows immediately that q(k) is invariantunder rotation of the orthonormal basis {ϕ1, ϕ2} in thelinear space spanned by them.

For the spectrally flatten Hamiltonian in Eq. (3), thewinding number over a given loop l is simply given by

w =1

2πi

∮l

dl Tr[q−1(k)∇lq(k)]. (16)

w exactly defines the topological charge of a point nodewhen the path l is chosen to be circling around this par-ticular node only. It should be noted that the integral inEq. (16) is taken to be counterclockwise along path l inthe main text.

2

B. Phase difference between s± and dx2−y2 pairing inhybrid Josephson junction

Consider an arbitrary phase difference θ between theiron-based SC (s± pairing) and cuprate SC (dx2−y2 pair-ing) in hybrid Josephson junction depicted in the maintext. Rashba layer sandwiched between the two SCswould develop a superconducting term with mixed pair-ing, given by

(∆s + eiθ∆d)(c†k↑c†−k↓ − c

†k↓c†−k↑) + H.c., (17)

where

∆s= ∆0 + 2∆1(cos kx + cos ky), (18)

∆d= 2∆2(cos kx − cos ky). (19)

We may perform a gauge transformation on the Nambuspinor basis, which send ck to

ck = cke−i 12 (θ−π2 ), (20)

and the pairing term in Eq. (17) could be rewritten as

[∆s sin θ + i(∆s cos θ + ∆d)](c†k↑c†−k↓ − c

†k↓c†−k↑) + H.c.

(21)Clearly, Eq. (21) exactly describes s+ i(s+ d) pairing asin the main text, where the first term ∆s sin θ is knownto be responsible for first-order or nodal superconductingphase, and the second one with a π/2-phase shift wouldopen a finite gap on each edge. Define

∆0 = ∆0 sin θ, ∆1 = ∆1 sin θ, ∆2 = (∆1 cos θ + ∆2)

δ =cos θ − β∆

cos θ + β∆

, η =2α∆ cos θ

cos θ + β∆

, (22)

with α∆ = ∆0/(4∆1), β∆ = ∆2/∆1, and we could thenrecover the same pairing form as in the main text, i.e.,

(∆s − i∆sd)(c†k↑c†−k↓ − c

†k↓c†−k↑) + H.c., (23)

with

∆s= ∆0 + 2∆1(cos kx + cos ky), (24)

∆sd= −2∆2(cos kx + δ cos ky + η). (25)

Hence, those criteria obtained before could be appliedstraightforwardly in this circumstance. The resultingcondition for the second-order topological phase is thengiven by

|µ− 4t∆| < 2√

2|A|√

1−∆2, θ 6= 0, π. (26)

The requirement of θ 6= 0, π in Eq. (26) guarantees thatthe pairing amplitude ∆s in Eq. (24) is nonzero. Also, weshould note that the inequality, |α∆| < 1, is always validfor s± pairing. This result implies that the criterion forsecond-order phase is independent of the phase differenceθ if only the latter doesn’t take 0 or π. Therefore, wedon’t require the phase difference of the hybrid Josephsonjunction to be fine-tuned to certain values. Instead, itcan take any value other than 0 and π, where the latteris known to be s± d pairing.

A subtle issue of Eq. (22) is that, δ and η thereinwould be ill defined when cos θ + β∆ = 0. In this case,the mass gaps of adjacent edges would be given by

msd= −4∆1 cos θ(cos kcy + α∆), (27)

m′sd= −4∆1 cos θ(− cos kcy − α∆), (28)

and hence we always have msdm′sd < 0, and the criterion

in Eq. (26) would still be valid.