Singlet-Quintet Mixing in Spin-Orbit Coupled Superconductors with Fermions
In non-centrosymmetric superconductors, spin-orbit coupling can induce an unconventional superconducting state with a mixture of s-wave spin-singlet and p-wave spin-triplet channelsBauer and Sigrist (2012); Gor’kov and Rashba (2001); Frigeri et al. (2004a), which leads to a variety of exotic phenomena, including anisotropic upper critical fieldBauer and Sigrist (2012); Yasuda et al. (2004); Takeuchi et al. (2006); Settai et al. (2008); Mukuda et al. (2010), magnetoelectric effectBauer and Sigrist (2012); Yip (2002); Edelstein (1995); Fujimoto (2005), topological superconductivitySato and Fujimoto (2009); Tanaka et al. (2009), et alBauer and Sigrist (2012). It is commonly thought that inversion symmetry breaking is substantial for pairing-mixed superconducting states. In this work, we theoretically propose that a new type of pairing-mixed state, namely the mixture of s-wave spin-singlet and d-wave spin-quintet channels, can be induced by spin-orbit coupling even in the presence of inversion symmetry when electrons effectively carry “spin-3/2” in superconductors. As a physical consequence of the singlet-quintet pairing mixing, topological nodal-line superconductivity is found in such system and gives rise to flat surface Majorana bands. Our work provides a possible explanation of unconventional superconducting behaviors observed in superconducting half-Heusler compoundsBrydon et al. (2016); Butch et al. (2011); Kim et al. (2016); Bay et al. (2012); Meinert (2016).
In the Bardeen-Cooper-Schrieffer theory, the s-wave spin-singlet pairing relies on the presence of both time reversal and inversion symmetry in superconductors (SCs). In non-centrosymmetric SCs, the absence of inversion symmetry can give rise to anti-symmetric spin-orbit coupling (SOC) with odd parity, and results in a mixture of s-wave spin-singlet (even parity) and p-wave spin-triplet (odd parity) pairings Bauer and Sigrist (2012); Gor’kov and Rashba (2001); Frigeri et al. (2004a). Due to the opposite parities of singlet and triplet pairings, only anti-symmetric SOC is considered in pairing mixing mechanism Bauer and Sigrist (2012), while symmetric SOC with even parity is normally overlooked in non-centrosymmetric SCs. However, we will show below this is not true if electrons carry “spin-3/2”. Here “spin” refers to total angular momentum , which is a combination of 1/2-spin and angular momentum of p atomic orbitals (), of basis electronic states. Such superconductivity with electrons was recently proposed in superconducting half-Heusler compounds Brydon et al. (2016), where unconventional superconducting behaviors, including low carrier density Butch et al. (2011); Kim et al. (2016); Bay et al. (2012); Meinert (2016), power-law temperature dependence of London penetration depth Kim et al. (2016) and large upper critical fieldBay et al. (2012), have been observed. Superconductivity with spin-3/2 fermions has also been considered in cold atom systemsWu (2006). In contrast to spin-1/2 SCs with only singlet and triplet states, the Cooper pairs of electrons can carry total spin (singlet), 1 (triplet), 2 (quintet) and 3 (septet). In this work, we demonstrate a new pairing-mixed state, namely the mixing between s-wave spin-singlet and d-wave spin-quintet pairings, can appear in spin-orbit coupled SCs with electrons, even in the presence of inversion symmetry. In particular, we will illustrate the role of symmetric SOC (parity-even) in the singlet-quintet mixing and how such pairing mixing can give rise to topological nodal-line superconductivity (TNLS).
We start from electronic band structures of half-Heusler compounds and illustrate the origin of electrons. The energy bands near the Fermi energy in half-Heusler compounds are s-type bands ( bands) and p-type bands, where the latter is split into bands ( bands) and bands ( bands) by SOC Winkler et al. (2003). For half-Heusler SCs with p-type of carriers like YPtBiButch et al. (2011), only the bands are relevantChadov et al. (2010), and can be described by four-component wavefunctions, labeled as , with total angular momentum that can be effectively regarded as “spin” and . The low energy physics of the bands is described by the so-called Luttinger modelChadov et al. (2010); Luttinger (1956) with the Hamiltonian
on the basis wavefunctions of , where with the chemical potential . The detailed forms of five d-orbital cubic harmonics ’s and six 4-by-4 matrices () are defined in Sec.A of supplementary materials (SMs). The above Hamiltonian only includes symmetric SOC term , while the antisymmetric SOC that breaks inversion will be discussed in the end. The Luttinger Hamiltonian is invariant if , and its symmetry is reduced to group if . The eigen-states of are doubly degnerate with eigen-energies , where the subscript labels two spin-split bands, and with , , and . We focus on the parameter regime with Yang et al. (2017a), (p-type carriers), and for simplicity. With the choice of these parameters, the effective mass of the band is always negative while there are three different regimes for of the band: (I) , (II) , and (III) the sign of being angular dependent. Energy dispersions and Fermi surface shapes in these three regimes are depicted in Fig.1a. In realistic materials, the regime I appears for the normal band structure when bands have higher energy than bands while the regime II exists for the inverted band structure with bands below bands.Chadov et al. (2010) In the regime III, the band disperses oppositely along the direction and , thus forming a saddle point at (Fig.1a(iii)) and hyperbolic Fermi surface (Fig.1a(vi)). We notice that in realistic materialsMeinert (2016); Yang et al. (2017a), the bands should eventually bend up at a large momentum in all directions (the dashed lines in Fig. 1a(iii) and (vi)). Thus, the Luttinger model is only valid in a small momentum region around in the regime III.
Next we will discuss the interaction Hamiltonian and the possible superconducting pairings in the Luttinger model. Several types of pairing forms have been discussed in literature, including mixed singlet-septet pairingBrydon et al. (2016); Kim et al. (2016); Yang et al. (2017b); Timm et al. (2017), s-wave quintet pairingBrydon et al. (2016); Roy et al. (2017); Timm et al. (2017); Boettcher and Herbut (2017) , d-wave quintet pairingYang et al. (2016); Venderbos et al. (2017) , odd-parity (triplet and septet) paringsYang et al. (2016); Venderbos et al. (2017); Savary et al. (2017), et alVenderbos et al. (2017). In particular, it is argued that s-wave singlet can be mixed with p-wave septet due to antisymmetric SOC Brydon et al. (2016); Kim et al. (2016). Here we focus on possible pairing mixing induced by symmetric SOC . In analog to the singlet-triplet mixing, in which the p-wave character of triplet channel originates from the p-wave nature of anti-symmetric SOC termFrigeri et al. (2004a), it is natural to expect that the pairing channel that is mixed into singlet channel due to should have d-wave nature with orbital angular momentum , given the d-wave in . According to the symmetry classification of gap functions for fermionsSavary et al. (2017) and the coupled linearized gap equations (See Sec.B4 of SMs), the only channel that can be mixed with s-wave singlet channel is d-wave quintet channel, which carries =(2,2,0) with spin =2 (quintet) and total angular momentum =0 () for the Cooper pair, under symmetry. Here we focus on a minimal -invariant interaction
in the s-wave singlet and d-wave quintet channels, where , , and and stand for the s-wave and d-wave interaction parameters, respectively. Here is the four-component creation operator on the basis , is the time-reversal matrix, is volume and is lattice constant. As discussed in Sec.B5 of SMs, the above interaction Hamiltonian can be extracted from the electron-optical phonon interaction proposed in Ref.Savary et al. (2017).
According to the interaction in Eq.2, the gap function should take the form , in which and represent s-wave singlet and d-wave quintet channels, respectively. The corresponding coupled linearized gap equation can be derived as (Sec.B6 of SMs)
where , is the Euler constant, is Boltzman constant, is the critical temperature, is the energy cut-off for the attractive interaction(), and are the normalized interaction parameters with the density of state , and and are the normalized order parameters. The band information is included in the functions . In the limit , and , the functions can be perturbatively expanded as , and up to the leading order, where means taking the real part, represents averaging over the solid angle , and are the normalized effective masses of the bands. As demonstrated in Sec.B6 of SMs, zero can lead to a vanishing off-diagonal term in the gap equation () due to , thus revealing the essential role of in singlet-quintet mixing.
By solving Eq. (3), the mixing ratio is evaluated numerically as a function of in Fig.1b (blue line) for and , which reveals different behaviors in three parameter regimes I, II and III. increases rapidly with in regime I, and diverges in regime III. The dominant d-wave quintet pairing in regime III originates from the faster divergence of compared to in Eq. (3). To take into account the limitation of the Luttinger model in parameter regime III, a momentum cut-off is introduced in computing as shown in Sec.B7 of SMs. With , a peak strucure of (the red line in Fig. 1b) is found and confirms the dominant role of d-wave quintet pairing in regime III. Other features of in the regime III (e.g. the kinks) are discussed in Sec.B7 of SMs. With further increasing (regime II), drops rapidly due to the disappearance of Fermi surface for the bands and thus simple s-wave singlet pairing dominates in this regime. In Fig.1c, the critical temperatures as a function of are revealed by the red line for the pairing mixing case, and by the orange and blue lines for the pure singlet and quintet cases, respectively. We find that (1) pairing mixing can help enhance critical temperature; and (2) singlet pairing dominates for most of regime I and the entire regime II while quintet pairing plays a vital role around regime III.
Similar to the singlet-triplet mixing in non-centrosymmetric SCsBauer and Sigrist (2012); Sato and Fujimoto (2009); Schnyder et al. (2012); Brydon et al. (2011); Yada et al. (2011), a physical consequence of singlet-quintet mixing is the existence of TNLS in certain parameter regimes. The topological property of superconducting phases can be extracted from the Bogoliubov-de Gennes Hamiltonian with the gap function determined by the gap equation (Eq. 3). TNLS can exist in the regime II when and and in the regime I and III as long as (Sec.C 2, 3, 5 and 7 of SMs). Here we focus on the regime I with normal band structure and . Fig.2a shows the phase diagram in the parameter space spanned by SOC strength and interaction strength ratio . Nodal rings are found in the yellow and red regions of Fig.2a for the band (Fig. 2b and e). Due to time reversal and inversion, a four-fold degeneracy exists at each point on the nodal ring. Fig. 2b (i-iv) reveals the evolution of nodal rings along the path depicted in the inset of Fig. 2a. Six nodal rings first emerge and center around the , and axes in Fig.2b (i). These nodal rings expand (Fig.2b (ii)) and touch each other, resulting in a Lifshitz transition (Fig.2b (iii)). After the transition, eight nodal rings with their centers at the and other three equivalent axes (Fig.2b (iv)) shrink to eight points and eventually disappear. Topological nature of these nodal rings can be extracted by evaluating topological invariant of one dimensional AIII class Schnyder and Ryu (2011) along the loop shown by the red circle in Fig.2b(i) (See Sec.C4 of SMs for detals). Direct calculation gives , coinciding with four-fold degeneracy of the nodal rings. Non-zero also implies the existence of Majorana flat bands at the surface of TNLS. Fig. 2c(More details in Sec.C8 of SMs) and d show the zero-energy density of states and the energy dispersions at the (111) surface, which are calculated from the iterative Green function method Sancho et al. (1985). The evolution of surface Majorana flat bands follows that of nodal ring structures (see Fig. 2c (i-iv) and d (i-iv)). Additional nodal rings exist in the red region of the phase diagram (Fig. 2a), as shown in Fig. 2e.
We finally discuss the experimental implications of our theory. Previous theoretical studies on half-Heusler SCs mainly focus on the compounds in regime II (inverted band structure), while our study suggests that regimes I (normal band structure) and III (a special case of inverted band structure) are more interesting due to strong singlet-quintet mixing. Superconductivity has been found in DyPdBi and YPdBi with normal band structure Nakajima et al. (2015) and critical temperatures around and , respectively, thus providing good candidates for TNLS. YPtBi is a SC with inverted band structureChadov et al. (2010) and recent first principles calculations Meinert (2016); Brydon et al. (2016); Yang et al. (2017a) suggest that its energy dispersion might belong to regime III, although debates still existBrydon et al. (2016); Kim et al. (2016). Evidence of TNLS has been found in the penetration depth experimentKim et al. (2016) . Previous study attributes the nodal structure to the p-wave septet pairing mixed with subdominant s-wave singlet pairing due to asymmetric SOCBrydon et al. (2016); Kim et al. (2016). Our work here provides an alternative explanation of the nodal structure as a result of singlet-quintet mixing induced by symmetric SOC . In realistic half-Heusler compounds, the energy scale of symmetric SOC () is two orders of magnitude larger than anti-symmetric SOC () Brydon et al. (2016); Savary et al. (2017). Thus, anti-symmetric SOC should be regarded as a perturbation and its influence on nodal-ring structures is discussed in Sec.C6 of SMs. Furthermore, the interaction in s-wave singlet channel is normally the dominant mechanism for superconductivity in weakly correlated materials. Therefore, we expect singlet-quintet mixing should be dominant over singlet-septet mixing and response for the nodal line structure in realistic SCs. Our new pairing mixing mechanism opens up a door to explore other exotic superconducting phenomena in spin-orbit coupled SCs with electrons.
JY owes a large amount of thanks to Lun-Hui Hu for patiently answering his questions on superconductivity. JY also thanks Rui-Xing Zhang, Yang Ge and Jian-Xiao Zhang for helpful discussion. CXL and JY acknowledge the support from Office of Naval Research (Grant No. N00014-15-1-2675).
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Appendix A Expressions and Conventions
The five d-orbital cubic harmonics are given by Murakami et al. (2004)
The angular momentum matrices of are written as Murakami et al. (2004)
The five Gamma matrices are defined as Murakami et al. (2004)
Clearly, where is the 4 by 4 identity matrix.
Time reversal matrix is defined as
where is the time-reversal operator. The convention of time reversal matrix chosen in this work is Savary et al. (2017)
where and is the anti-commutator.Murakami et al. (2004)
is defined to satisfy
for any three dimensional(3d) unit vector and any angle , where are the angular momentum matrices on the bases of the spin tensors.
Since the spin tensor is a rank-2 tensor, it can be viewed as the addition of two copies of spin basis. In the spin- case, there are spin tensors with ranging from to and ranging from to . The chosen expressions in this work are shown as the followingYang et al. (2016); Savary et al. (2017):
and . The spin tensors satisfy the orthogonal condition .
Furthermore, matrices satisfy the relation
Appendix B Linearized gap equation and singlet-quintet mixing in Luttinger model
b.1 Green Functions of Luttinger model
The Luttinger model shown in the main text can be rewritten as
where , and . That gives
and eigenenergies of are
The Green functions of the Luttinger model are given by
for electrons and holes, respectively. Here we use the fact that is time-reversal invariant.
The Green functions can also be expressed in terms of projection operators , defined as
in the subspace of the bands, where stands for the double degeneracy of each band. In the chosen bases, the matrix forms of are
with . Correspondingly,
The isotropic case corresponds in the above expressions. Since commutes with for , energy eigenstates can be labeled with eigenvalues of . In this case, the bands are bands if , and bands if .
b.2 Expansion of interaction and gap function into different Channels
This part follows Ref.Savary et al. (2017). Consider a three dimensional density-density interaction
After performing the Fourier transformation, we obtain
with the total volume .
Since Cooper pairs of superconductivity occurs for two electrons with opposite momenta, we only keep the terms with in the above interaction. As a result, we can define and , which lead to
We generally denote as and impose the symmetry on the interaction, for any . In addition, the Hermitian condition of interaction requires .
Due to the symmetry, can be expanded as
Here the spherical harmonic functions satisfy the orthogonal condition .
Since both and form irreducible representations (irreps) of group, their product can be decomposed into new irreps with ClebschâGordan(C-G) coefficients as
where can be easily derived from the orthogonal conditions of ’s and ’s.
With the above expansion, we have
which gives rise to
Due to the anti-commutation relation of fermion operators, we have
which gives a constraint on the form of . Since , it requires to be an even number. As a summary, the form of interaction term is given by