This is the submitted version of the article: Ghosh S.K., Csire G., Whittlesea P., Annett J.F., Gradhand M., Újfalussy B., Quintanilla J.. Quantitative theory of triplet pairing in the unconventional superconductor LaNiGa2. Physical Review B, (2020). 101. 100506: - . 10.1103/PhysRevB.101.100506. Available at: https://dx.doi.org/10.1103/PhysRevB.101.100506 Quantitative Theory of Triplet Pairing in the Unconventional Superconductor LaNiGa2 Sudeep Kumar Ghosh,1, ∗ Gábor Csire,2, 3, † Philip Whittlesea,1 James F. Annett,2 Martin Gradhand,2 Balázs Újfalussy,4 and Jorge Quintanilla1, ‡ 1 Physics of Quantum Materials, School of Physical Sciences, University of Kent, Canterbury CT2 7NH, United Kingdom 2 H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom 3 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC, BIST, Campus UAB, Bellaterra, Barcelona, 08193, Spain 4 Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, PO Box 49, H-1525 Budapest, Hungary (Dated: March 16, 2020) The exceptionally low-symmetry crystal structures of the time-reversal symmetry breaking su- perconductors LaNiC2 and LaNiGa2 lead to an internally-antisymmetric non-unitary triplet (INT) state as the only possibility compatible with experiments. We argue that this state has a distinct signature: a double-peak structure in the Density of States (DOS) which resolves in the spin channel in a particular way. We construct a detailed model of LaNiGa2 capturing its electronic band struc- ture and magnetic properties ab initio. The pairing mechanism is described via a single adjustable parameter. The latter is fixed by the critical temperature Tc allowing parameter-free predictions. We compute the electronic specific heat and find excellent agreement with experiment. The size of the ordered moment in the superconducting state is compatible with zero-field muon spin relaxation experiments and the predicted spin-resolved DOS suggests the spin-splitting is within the reach of present experimental technology. The superconducting state is a condensate of electron the point group of the crystal. As an example, the point pairs characterized by an order parameter ∆. Usually ∆ group of Sr2 RuO4 is D4h , which leads to 22 possible order is a complex scalar, its phase being a manifestation of parameters breaking TRS [23, 24]: 20 under the assump- spontaneously-broken gauge symmetry. This is responsi- tion of weak spin-orbit coupling (SOC) and two more in ble for the macroscopic quantum coherence underpinning the strong-SOC limit. The family formed by LaNiC2 [9] quantum devices such as superconducting qubits [1] and and LaNiGa2 [10] are an exception to this rule, as SQUIDs [2]. On the other hand, in so-called “unconven- their crystal structures have exceptionally low symme- tional” superconductors additional symmetries may be try. Their crystal point groups only have four irreps, all broken leading to more complex order parameters with of them one-dimensional. This precludes TRS breaking extra degrees of freedom. Of all the features of uncon- in the strong-SOC case and leaves only four possible pair- ventional superconductors, broken time-reversal symme- ing states, all of them non-unitary triplets [10, 25]. One try (TRS) is perhaps the most surprising one as it chal- additional complication is the multi-band nature of these lenges our view of superconductivity and magnetism as systems: two [26] and five [27] bands cross the Fermi level antagonistic states of matter. In spite of this, the phe- of LaNiC2 and LaNiGa2 , respectively. In fact, both sys- nomenon has been detected in numerous systems using tems show thermodynamic properties that can be fitted zero-field muon spin rotation/relaxation (µSR). Promi- with a model assuming fully-gapped, 2-band supercon- nent examples include (U, Th)Be13 [3], Sr2 RuO4 [4], ductivity [11, 28, 29]. This is inconsistent with the line UPt3 [5], (Pr, La)(Ru, Os)4 Sb12 [6, 7], PrPt4 Ge12 [8], nodes implied by the earlier symmetry analyses [10, 25]. LaNiC2 [9], LaNiGa2 [10, 11], SrPtAs [12], Re6 (Zr, Hf, On the other hand the 2-band model does not predict Ti) [13–16], Lu5 Rh6 Sn18 [17] and La7 (Ir, Rh)3 [18, 19]. TRS breaking. To resolve the discrepancy it was pro- Many of these systems have other unconventional fea- posed that only an internally-antisymmetric non-unitary tures, while in some cases an independent, direct obser- triplet pairing (INT) state is compatible with the exper- vation of broken TRS has been made: optical Kerr effect imental observations [10, 11, 25, 30]. Here we show that measurement in Sr2 RuO4 [20] and UPt3 [21], and bulk such a state has a very distinct experimental signature: SQUID magnetization measurement in LaNiC2 [22]. a double-peak structure in the Density of States (DOS) which resolves in the spin channel. We construct a model Unfortunately it has been difficult to establish the of LaNiGa2 capturing detailed electronic band structure structures of order parameters of these superconductors. ab initio, with the pairing interaction in the INT state This is because, on the one hand, our knowledge of the reduced to a single, adjustable parameter. The known electron pairing mechanism is not sufficiently detailed to value of the critical temperature Tc fixes this single pa- make a prediction. On the other hand, their crystal struc- rameter, allowing us to make parameter-free predictions. tures tend to be highly symmetric, leading to many differ- We obtain the electronic specific heat and find an excel- ent possible ways of breaking TRS, which limits our abil- lent agreement with experiment [11]. We compute the ity to work by elimination. TRS-breaking superconduc- spin-resolved DOS having a double-peak structure with tivity requires a degenerate instability channel [23, 24] each peak corresponding to a single spin channel. We which, for a uniform superconductor, must come from find that the splitting ∼ 0.2meV —within the reach of a multi-dimensional irreducible representation (irrep) of 2 present experimental technology. 0.6 ↑ The triplet pairing in the INT state relies heavily on 1.5 ↓ the inter-band pairing, which enables an isotropic gap 0.4 total DOS 1 function and equal-spin pairing breaking TRS [11]. The Cooper pair wave function is symmetric in the crystal 0.2 0.5 momentum and spin channels but it is anti-symmetric ↑, + ↑, - (b) σ = 0.07 with respect to the orbital degree of freedom. Recent Ek 0 0 studies [11, 31–42] in several materials, including the Iron ↓, + 0.8 ↓, - based superconductors, half-Heusler compounds, UPt3 -0.5 0.6 and Sr2 RuO4 , have also pointed out the importance of DOS internal degrees of freedom of electrons (coming from, for -1 0.4 example, sublattice or multiple orbitals) in determining the pairing symmetries of superconducting ground states. -1.5 0.2 (a) (c) σ = 0.025 A convenient toy model of low-energy excitations in 0 -1 -0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 the INT state proposed in Ref. [11] is provided by the ǫ0(k) - µ Ek following Bogoliubov-de Gennes (BdG) Hamiltonian: ˆ   H0 (k) ∆ FIG. 1. Properties of quasiparticles in H= (1) pthe INT state with s = ˆ † −H0 (k) . ∆ 0.05, δ = 0.075, |∆0 | = 1 and |q| = 2/3 in arbitrary units. (a) Quasiparticle spectrum for the + and − bands for ↑ and Here k is the crystal momentum of the excitation, ↓ spins. (b) and (c) show the corresponding DOS calculated   from this spectrum using the same parameters. The DOS  (k) − µ − s δ features have been artificially broadened by convolution with H0 (k) = 12 ⊗ 0 (2) a Gaussian of width σ = 0.07 in (b) and σ = 0.025 in (c). δ 0 (k) − µ + s is the normal-state, single-electron Hamiltonian with the chemical potential µ and four pairs of coherence peaks that are grouped in two doublets, depending on the level of broadening. Cru- ˆ = i(d.σ ∆ σ )σy ⊗ iτy (3) cially, the spin-resolved DOS shows only one of the two doublets in each spin channel. This qualitative feature represents the pairing potential. In the tensor products, distinguishes this double-peak structure from that which the first sector represents the spin channels σ =↑, ↓ while would be obtained, for example, in a multi-band super- the second represents the two orbital channels. For the conductor. The observation of such a spin-resolved fea- purpose of initial discussion, we have assumed a very ture would provide definitive proof of the INT state. simple band structure with two bands labeled by m = + An analytical formula for Ek can be easily obtained in and −, one emerging from each orbital, that are related the limits s → 0 or δ → 0. In either case, the result is by a rigid energy shift 2s and with a k−independent hybridization factor δ. The pairing matrix describes h p i Ek = ± ±a + {0 (k) − µ}2 + |∆0 |2 (1 ± |q|) , (4) k−independent triplet pairing but is antisymmetric in the orbital channel in order to ensure the fermionic an- where a = δ or s, respectively. Note that both δ and s tisymmetry of the Cooper pair wave function [11, 31]. play similar roles. This shows that our toy model does Here, σ and τ are the vectors of Pauli matrices in the not rely on the two bands being orthogonal. The above spin and orbital sectors respectively. Writing the triplet formula can be used to estimate the ratio between the d-vector in the form d = ∆0η , where |ηη |2 = 1 and ∆0 is gaps in the energy spectrum for spin-up (E↑↑ ) and spin- a pairing amplitude, the nonunitarity of the triplet state down (E↓↓ ) quasiparticles. In the limit a  ∆0 , it is is characterized by a nonzero real vector q = i(ηη × η ∗ ) which in general has |q| ≤ |ηη |2 = 1. E↑↑ p 1 + |q| Diagonalizing H yields the quasi-particle spectrum Ek =p = 1 − |q| + O(|q|2 ). (5) E↓↓ 1 − |q| shown, for a particular choice of parameters, in Fig. 1(a). The plot is representative of cases where s, δ  ∆0 . This The above toy model assumes that an isotropic, equal- is the physically-relevant regime for the toy model as in spin pairing potential can lower the free energy in spite a mean-field picture the pairing amplitude (∆0 ) has to of the need for it to breach the energy gap between the be able to overcome the band splitting ∼ δ, s. This un- bands. We explicitly show this by considering the toy realistic requirement is relaxed when the band splitting many-body Hamiltonian is allowed to be k-dependent, as in the more detailed model discussed below. As indicated in the plot each X † Ĥ = Ψ̂k H0 (k)Ψ̂k + ĤI . (6) excitation has well-defined band and spin indices. The k Bogoliubov bands are paired up, with each member of the pair sharing the spin index but differing in the band Here Ψk = (ĉk,+,↑ , ĉk,+,↓ , ĉk,−,↑ , ĉk,−,↓ ), where ĉk,m,σ index. The corresponding DOS is displayed in Fig. 1 (b) creates an electron in the mth band with crystal mo- and (c). Here we have introduced two different levels mentum k and spin σ. The single-electron Hamiltonian of broadening to simulate different experimental resolu- H0 (k) is given in Eq. (2) where for simplicity we take tions in the two figures. The DOS is fully-gapped, with δ = 0 and 0 (k) = −2t[cos(kx )+cos(ky )]. We consider an 3 on-site, inter-orbital, equal-spin pairing interaction pro- band superconductor with several bands crossing the posed in Ref. [11], which can be written [43] as Fermi level giving rise to multiple Fermi surface X † sheets [27]. None of the bands can be obtained from ĤI = −U ck,+,σ c†−k,−,σ c−k0 ,−,σ ck0 ,+,σ (7) one another through a simple rigid shift as in our toy k,k0 ,σ model. There are, however, several regions within the Brillouin zone where the pairs of Fermi surface sheets with U > 0 being the effective attraction strength. A are parallel and very close to each other, that is, nearly standard mean-field treatment of this model [44] yields degenerate [44]. Moreover, the five bands have mixed Ni the phase diagram shown in Fig. 2. In the limit s → 0, 3d, La 5d and Ga 4p characters. As a result, the Fermi the theory is formally equivalent to two copies of a BCS surfaces have strong orbital degeneracy. To capture these theory, but with the band index m playing the role of the details, we adopt a semi-phenomenological strategy [46]. spin index (one copy corresponding to each value of the We consider the relativistic version of the BdG Hamil- real spin). For finite s, a critical interaction strength Uc is tonian in Eq. (1) together with the realistic LDA band necessary for the critical temperature Tc to be finite, but structure and a phenomenological pairing interaction of for U  Uc the results are very similar to the case s → 0. the type given by Eq. (7). This leads to the Kohn-Sham- This is confirmed by the inset, showing the temperature- Dirac-BdG Hamiltonian [47] dependence of the pairing amplitude ∆0 . While the su- perconducting transition can be of first order and even  ˆ  HD ∆(r) re-entrant (not shown; see Ref. [44]) for a very narrow HDBdG = ˆ (r) −H ∗ † (8) window U & Uc , and displays some shoulders for slightly ∆ D larger U , BCS-like behavior is recovered for U  Uc . where HD is the effective normal state Dirac Hamiltonian given by 0.4 U/t = 2.90 = 2.80 0.2 0.3 = 2.70 HD = cpα̂ α2 −14 )c2 /2+(Vef f (r)−EF )14 +Bef f (r)α̂ α1 +(α̂ α3 . = 2.66 (9) ∆0 / t 0.2 = 2.65 Here, α̂α1 = σ̂x ⊗ σ̂ α2 = σ̂z ⊗ 12 and α̂ σ , α̂ α3 = 12 ⊗ σ̂ σ with σ̂σ 0.1 being the Pauli matrices and 1n being the identity ma- kBTc / t trix of order n. Vef f (r) and Bef f (r) are the effective 0 0.1 0 0.05 0.1 0.15 0.2 electrostatic potential and the effective exchange-field, s / t = 0.000 kBT / t ˆ respectively. ∆(r) is the 4 × 4 pairing potential ma- = 0.025 trix due to the four component Dirac spinors. Requiring = 0.050 = 0.075 self-consistency in the electrostatic potential, exchange- = 0.100 field and pairing potential, the solution is provided by 0 our recently developed method [47] which generalizes the 0 0.5 1 1.5 2 2.5 3 U/t Korringa-Kohn-Rostoker (KKR) formalism. Within the KKR formalism intra-orbital and inter-orbital pairings could be described both in the singlet and triplet channel FIG. 2. Superconducting phase diagram of the toy model in by transforming the (L, σ) representation of the pairing Eq. (6). Tc is the critical temperature where inter-orbital, potential into a relativistic basis set, where L refers to the equal-spin pairing sets in. Each curve shows the dependence real spherical harmonics assuming that the z direction is of Tc on the interaction strength U for a different value of the band splitting 2s, as indicated. The inset shows the perpendicular to the layered structure of LaNiGa2 . The temperature-dependence of the pairing amplitude ∆0 for the technical details are given in the Ref. [44]. largest splitting s/t = 0.1 and a few values of U just above It is important to note that the ground state does not the critical value Uc at which Tc becomes finite. show ferromagnetism in the normal state, although there is a significant contribution from the Ni 3d states at the The above simple calculation shows that an equal-spin Fermi level [27]. Since it is known that Hund’s rule cou- pairing potential can, in principle, breach a band gap to pling plays an important role on the Ni atoms [48, 49], lead to a fully-gapped triplet pairing state. On the other and on the other hand Hund’s coupling can also pro- hand our simple mean field theory yields η = (0, 1, 0), i.e. duce local pairing [50], therefore, it is physically rea- a unitary triplet pairing state with q = 0. A more re- sonable to assume an inter-orbital equal-spin pairing in- alistic theory must treat the pairing and exchange fields volving two orbitals on the same Ni atom. We describe on equal footing. Symmetry arguments [10] show that this by a two-body onsite attractive interaction UL,L0 be- any triplet instability leads to a subdominant magneti- tween electrons with equal spins in only two of these or- sation, which lowers the free energy of the non-unitary bitals (L 6= L0 ) with the pairing potential satisfying the state with |q| = 6 0 [10, 45]. We now therefore build a more self-consistency equation: ∆Lσ,L0 σ (r) = UL,L0 χLσ,L0 σ (r) sophisticated, realistic description which not only incor- where χLσ,L0 σ (r) is the corresponding pairing amplitude. porates an accurate description of the exchange field, but Since all of the Ni d orbitals contribute to the density also a realistic prediction of the normal-state electronic of states at the Fermi level, there are 10 possible pair- structure of LaNiGa2 , making quantitative predictions of ing models within this approach. Only one of the 10 superconducting properties possible. possible combinations, namely pairing between dz2 and Density-Functional Theory (DFT) in the Local-density dxy , yields a fully-gapped quasi-particle spectrum (all the approximation (LDA) shows that LaNiGa2 is a multi- other possibilities have nodes on at least one of the Fermi 4 2 0.012 8 ↑ ↓ 1.5 0.009 6 total C / (γ T) µs ( µB ) DOS 1 0.006 4 Theory 0.5 Experiment 0.003 2 (a) (b) (c) 0 0 0 0 0.25 0.5 0.75 1 1.25 0 0.25 0.5 0.75 1 1.25 -1.5 -1 -0.5 0 0.5 1 1.5 T/Tc T/Tc Energy (meV) FIG. 3. (Color online) Properties of the superconducting ground state corresponding to a phenomenological inter-orbital equal- spin pairing interaction between the (dz2 - dxy ) orbitals in the 3d sector of the Ni atom in LaNiGa2 . (a) Variation of the specific heat with temperature (γ is the Sommerfeld coefficient). We note an excellent agreement between the theoretical result and the experimental data taken from Ref. [11]. (b) Variation of the spontaneous magnetic moment (µs ) as a function of temperature. A clear increase in the magnetic moment below Tc is an indication of the imbalance between two spin-species due to migration of Cooper pairs. (c) Spin-resolved density of states (arbitrary units) of the Bogoliubov quasi-particles as a function of energy. We note that the two coherence peaks correspond to up and down species of Cooper pairs leading to two gaps in the quasi-particle spectrum. µ µ0 surface sheets– see the Ref. [44]). The strength Udz2 ,dxy the value of µs at T = 0, as Bint ≈ 4πabc 0 s ≈ 0.3 Gauss of the interaction between these two orbitals is the only which is of similar order as seen in the zero-field µSR adjustable parameter in our theory, to be fixed by requir- measurements [10]. ing Tc to be the same as in experiments [11]. Then we Finally, we compute the spin-resolved quasiparticle can make parameter-free predictions of observable prop- DOS of LaNiGa2 as shown in Fig. 3(c). Its similarity erties of the system. The requirement that Tc is the same with Fig. 1(b) is striking, confirming that our DFT-KKR as in experiments leads to Udz2 ,dxy = 0.65 eV which is calculation for this material describes the same physics. comparable to the values of Hund’s coupling found for The two distinct superconducting gaps are clearly visible, Ni atom [49]. This result should motivate high-pressure and the spin-resolved curves show that they correspond measurements and Dynamical Mean Field Theory stud- to different spin species. The double-peak structure of ies to further explore the role of electronic correlations the DOS is our main prediction. We note that the split- involving Hund’s coupling. We stress that the present ting between the two peaks is of the order of 0.2 meV attractive interaction is described by a phenomenological —within the resolution of current scanning tunneling mi- parameter and therefore our calculation cannot directly croscopy [52], photo-emission [53], and tunneling exper- address the question of its origin. iments [54]. The crucial feature is that, unlike the case Having fixed our single parameter, we can now make of a multi-band superconductor [55], the two peaks cor- parameter-free predictions. We first compute the specific respond to distinct spin channels. Verifying this experi- heat of LaNiGa2 as a function of temperature by evalu- mentally would thus require spin resolution [56]. ating the temperature dependence of the quasi-particle Interestingly, in Ref. 11 the specific heat measurement DOS self-consistently [46, 51]. It is shown in Fig. 3(a) was fitted by a phenomenological two-band model lead- comparing to the corresponding experimental data from ing to the gap values ∆1 = 1.08kB Tc and ∆2 = 2.06kB Tc , Ref. [11]. The agreement is excellent, suggesting that while we find that the difference between the gaps is only the observed two-gap behavior of this curve is consistent around 20%. Clearly, the difference between the two with our equal-spin, inter-orbital pairing model. procedures comes from the fact that our first-principles The solution of the self-consistency equations reveal based calculation included all of the five bands crossing a charge imbalance between ↑↑ (67%) and ↓↓ (33%) the level. However, our main point is that the DOS is triplet components on the Ni atom. The migration of spin-polarized around the Fermi level, and the supercon- Cooper-pairs from the minority ↓↓ state to the majority ducting gaps correspond to different spin channels, not ↑↑ state is expected to generate a finite magnetization. different bands. Since our pairing interaction Udz2 ,dxy is spin indepen- We note that it is important to consider the effect of dent (hence preserves TRS), we have found a spontaneous magnetic and nonmagnetic impurities on the INT state. TRS breaking in the INT state, which is a perfect ana- Although this is outside the scope of the present paper, logue of a ferromagnetic transition in a normal-state DFT due to the two full gaps we expect the INT state to be calculation. The pairing-induced spontaneous magneti- protected from nonmagnetic impurity scattering and a zation (µs ) is shown in Fig. 3(b). The magnetic moment version of the Anderson’s theorem [57] to hold. is expected to vary linearly very close to Tc [10], however Conclusions: We showed that an unconventional su- this behavior is hard to resolve here due to demanding perconductor in the INT state has at least two gaps, one numerical accuracy near Tc . We can estimate the size of for each spin flavour, irrespective of the number of Fermi the internal magnetic field at zero temperature using µ0s , surfaces. Instead of the traditional route of ignoring the 5 microscopic complexity of Fermi surfaces, we consider the spin relaxation in UPt3 ,” Phys. Rev. Lett. 71, 1466–1469 fully-relativistic electronic band structure of LaNiGa2 . (1993). We perform fully self-consistent computations of its ob- [6] Y. Aoki, A. Tsuchiya, T. Kanayama, S. R. Saha, servables by taking a phenomenological pairing model on H. Sugawara, H. Sato, W. Higemoto, A. Koda, K. Ohishi, K. Nishiyama, and R. Kadono, “Time- the Ni atom in the INT state. 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