On the possibility of mixed helical p-wave pairings in Sr_2RuO_4
OOn the possibility of mixed helical p-wave pairings in Sr RuO Wen Huang ∗ and Zhiqiang Wang † Shenzhen Institute for Quantum Science and Engineering & Guangdong Provincial Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology, Shenzhen 518055, Guangdong, China James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA (Dated: February 9, 2021)The exact nature of the unconventional superconductivity in Sr RuO remains a mystery. At the phenomeno-logical level, no superconducting order parameter proposed thus far seems able to coherently account for allof the essential experimental signatures. One prominent observation is the polar Kerr effect, which implies ananomalous Hall effect. Without appealing to extrinsic factors originating from crystalline imperfection, thispeculiar effect may be interpreted as an evidence of a bulk chiral Cooper pairing with nonzero orbital angularmomentum, such as p + ip or d + id . In this paper, we propose alternative possibilities with complex mixturesof distinct helical p-wave order parameters, namely, A u + iA u and B u + iB u in the group theory nomencla-ture. These states essentially consist of two copies of inequivalent chiral p-wave pairings with opposite chiralityand different pairing amplitudes, and therefore support intrinsic Hall and Kerr effects. We further show thatthese states exhibit salient features which connect to several other key observations in this material, includingthe absence of spontaneous current, a substantial Knight shift drop, and possibly signatures in uniaxial strainand ultrasound measurements. The nature of the unconventional superconductivity inSr RuO is an outstanding puzzle [1–10]. Its normal stateproperties have been characterized with unprecedented accu-racy — an incredible feat that is not often seen in other un-conventional superconductors. The electron correlation in thismaterial is moderate, and superconductivity emerges out of awell-behaved Fermi liquid [3]. These have led to a popularperception that resolving its pairing mechanism and identify-ing its pairing symmetry should be well within the capacity ofestablished theories and experimental techniques. Yet, despitetremendous efforts over the past twenty plus years, neither ofthese two key issues has been settled. This is largely due tothe lack of an order parameter that could coherently interpretall the essential experimental observations [9, 10].Arguably, one principal property of the superconductivityin Sr RuO is the condensation of at least two supercon-ducting order parameters. This has been inferred from a va-riety of experiments, including µ SR [11, 12], optical polarKerr effect [13], Josephson interferometry [14], and ultra-sound [15, 16], etc. A multi-component pairing is realizedif the Cooper pairing is developed in a multi-dimensional ir-reducible representation (irrep) of the crystal point symme-try group. One notable example is the chiral p-wave pair-ing, i.e. ( k x + ik y )ˆ z , which belongs to the E u irrep of the D h group. Here, ˆ z denotes the orientation of the so-called d -vector, which describes the spin configuration of a spin-triplet pairing [3]. Another possibility is coexisting order pa-rameters associated with distinct one-dimensional irreps. Inthis scenario, the multiple components typically condense be-low distinct critical temperatures, although accidental near-degeneracy is possible, in principle.In light of the evidence of time-reversal symmetry breakingfrom the µ SR and the polar Kerr effect measurements [11–13], the multiple components presumably form a complexpairing, such that the state does not return to itself after atime-reversal operation. The Kerr effect, and the closely re-
Figure 1. (color online) Illustration of the Cooper pairing in themixed helical p-wave A u + iA u state in a single-orbital model.The state consists of two copies of chiral p-wave model with oppo-site chirality. The Cooper pairs are drawn with different size to reflecttheir different pairing amplitudes. Green arrows indicate the direc-tion of the Cooper pair orbital angular momentum L z . In the limit ∆ A u = ∆ A u , the state resembles the He A -phase. Another can-didate state B u + iB u has a similar pairing configuration. lated anomalous Hall effect, intuitively imply an underlyingchiral pairing with nonzero orbital angular momentum, suchas p + ip and d + id , as is substantiated by a recent study [17].Hence, if the Kerr effect in this material is intrinsic (i.e. notrelated to extrinsic effects such as impurities or lattice dis-locations), its superconducting symmetry is limited to a fewcandidates.In this work, we turn to the helical p-wave pairings ( A u , A u , B u and B u irreps), which are analogues of the He B-phase [18]. They individually preserve time-reversal symme-try, as each effectively consists of a pair of sub-Hilbert spacecharacterized respectively by k x + ik y and k x − ik y pairingswith identical gap amplitude. However, appropriate complexmixture of these helical pairings could tune the relative gapamplitude between k x + ik y and k x − ik y away from unity(see Fig. 1). The resultant ‘asymmetric’ state then supportsintrinsic Kerr effect. We are mainly interested in the scenar-ios of two-component order parameters, for which we identifytwo possibilities that meet our requirement: A u + iA u and B u + iB u . The idea is most transparent when presentedin a single-orbital model. Take the former as an example, in a r X i v : . [ c ond - m a t . s up r- c on ] F e b terms of the d -vector, the pairing functions of A u and A u are given respectively by k x ˆ x + k y ˆ y and k x ˆ y − k y ˆ x . The fullgap function then becomes, d k = ∆ A u ( k x ˆ x + k y ˆ y ) + i ∆ A u ( k x ˆ y − k y ˆ x ) , (1)where ∆ A u are real gap amplitudes. d k is non-unitarysince d ∗ k × d k = 0 [18], implying unequal pairing gap am-plitudes for the two spin species: ∆ ↑↑ ( k ) = − (∆ A u − ∆ A u )( k x − ik y ) for spin-up and ∆ ↓↓ ( k ) = (∆ A u +∆ A u )( k x + ik y ) for spin-down (Fig. 1). If ∆ A u = ∆ A u ,the state resembles the He A -phase which has one spin com-ponent unpaired [18]. More generally, both spin species arepaired and they carry opposite orbital chiralities, i.e. opposite L z . Such a state is odd-parity by nature [21], and it exhibitsother salient features potentially consistent with the absenceof spontaneous current signature [23, 24] and the substantialKnight shift drop [19, 20], as we shall see later. Since the B u + iB u state essentially shares the same general features,we will not present further analyses about it in the remainingof the paper.At this point, it is worth pointing out that multiple micro-scopic calculations [26–32] have found that helical pairingstates are favored in certain reasonable interaction parameterspace. More interestingly, in some cases the splitting amongvarious helical states could be rather small [32], making it rea-sonable to consider accidentally degenerate pairing. Here, wemake no attempt to delve upon the pairing mechanism or uponhow the pairing is distributed among the multiple orbitals inSr RuO . Instead, we will focus on the general phenomenathat do not rely on microscopic details. Apart from some occa-sional general discussions, our analyses will be largely basedon a concrete two-orbital model containing the Ru d xz and d yz orbitals. As far as the physical properties to be discussed areconcerned, we expect the same qualitative features for moregeneral multi-orbital models of Sr RuO . For simplicity, weassume the simple pairing functions for the various odd-parity(p-wave) superconducting channels as given in Table I. Table I. Representative simple odd-parity pairing gap functions ( d -vectors) for the two-orbital model containing the Ru d xz and d yz orbitals. irrep d xz d yz A u sin k x ˆ x sin k y ˆ yA u sin k x ˆ y − sin k y ˆ xB u sin k x ˆ x − sin k y ˆ yB u sin k x ˆ y sin k y ˆ xE u sin k x ˆ z sin k y ˆ z AC ANOMALOUS HALL EFFECT AND KERR ROTATION
Since our proposed state is essentially composed of a pairof (spinless) chiral p-wave models with opposite chirality andinequivalent gap amplitudes, we expect it to support anoma-lous Hall effect. Notably, while the intrinsic anomalous Hall effect is generally absent in single-orbital chiral superconduct-ing models [33–36], Sr RuO has an inherent multi-orbitalcharacter.Below we demonstrate intrinsic a.c. anomalous Hall effectand Kerr rotation for the A u + iA u state, using the sametwo-orbital model as in Ref. [34]. Without qualitative im-pacts on our results, we neglect spin-orbit coupling (SOC) inthis section for simplicity. Then the spin up and down blocksdecouple and their contributions to the two-dimensional Hallconductivity σ H can be computed separately. Here, we brieflydescribe the spin up block of the model. Written in the basis ( c d xz , ↑ ( k ) , c d yz , ↑ ( k )) , where c d xz , ↑ and c d yz , ↑ are annihila-tion operators for the d xz and d yz orbitals of Sr RuO , thenormal state Hamiltonian reads, H N ( k ) = (cid:18) ξ xz ( k ) λ ( k ) λ ( k ) ξ yz ( k ) (cid:19) , (2)where ξ xz ( k ) = − t cos k x − t cos( k y ) − µ , ξ yz ( k ) = − t cos k x − t cos( k y ) − µ , λ ( k ) = 4 t sin k x sin k y . Here t labels the nearest neighbor hopping integral of the d xz ( yz ) orbital along the x ( y ) direction, ˜ t that of the d xz ( yz ) orbitalalong the y ( x ) direction, t denotes the hybridization of thetwo orbitals between next nearest neighboring sites. The cor-responding BdG Hamiltonian follows as, H ↑ BdG ( k ) = H N ( k ) ˆ∆ ↑↑ ( k )ˆ∆ †↑↑ ( k ) − H T N ( − k ) ! , (3)where, following Table I, ˆ∆ ↑↑ ( k ) = − (∆ A u − ∆ A u ) (cid:18) sin k x − i sin k y (cid:19) . (4)Using H ↑ BdG we calculate σ ↑ H ( ω ) from the standard Kuboformula (without vertex corrections). Details of the calcula-tion can be found in the Supplementary Materials [37]. Thesame calculation is done for σ ↓ H and the numerical result of thetotal σ H at temperature T = 0 is presented in Fig. 2. Not sur-prisingly, σ H is nonzero; both its real and imaginary parts havesimilar frequency dependence to those obtained for a chiral p-wave pairing [34]. The overall magnitude of σ H is reduced bya factor of two for equal or comparable magnitudes of ∆ A u and ∆ A u , compared to that in Ref. [34], because in the cur-rent scenario σ ↑ H and σ ↓ H partially cancel out each other whilethe cancellation is absent in the chiral p-wave pairing case.We note that Im[ σ H ( ω )] becomes nonzero at ω & . for the chosen band parameters. This frequency is determinedby the minimum energy cost to create a pair of Bogoliubovquasiparticles from different bands, which is not dictated by | ∆ ↑↑ | or | ∆ ↓↓ | but by the hopping parameter t [34]. There-fore, even though | ∆ ↑↑ | and | ∆ ↓↓ | can be quite different, theonset frequencies of Im[ σ ↑ H ] and Im[ σ ↓ H ] are actually almostidentical.From the calculated σ H one can estimate the Kerr rotationangle using θ K ( ω ) = 4 πωd Im[ σ H ( ω ) n ( n −
1) ] , (5) × - × - × - ℏω [ eV ] σ H [ e / ℏ ] Figure 2. (color online) Real (blue solid line) and imaginary (reddashed line) parts of the σ H ( ω ) as a function of ω at T = 0 . Bandparameters are [34]: t = µ = 0 . , t = 0 . t and ˜ t = 0 . t . ∆ A u = 2∆ A u = ∆ max with ∆ max = | ∆ A u | + | ∆ A u | =0 .
23 meV [34]. We choose | ∆ A u | 6 = | ∆ A u | so that the gap mag-nitudes are nonzero for both spin up and down; otherwise, one spincomponent would remain normal at zero temperature which is notobserved. where n = n ( ω ) is the complex index of refraction and d isthe interlayer spacing of Sr RuO along the c -axis. n ( ω ) canbe estimated from the longitudinal optical conductivity whichis modeled by a Drude model as in Ref. [34]. For details seeRef. 37. The estimated θ K ( ~ ω = 0 . eV ) ≈
20 nrad [37],which may potentially account for the experimental value of
65 nrad observed by J. Xia et al [38], given the uncertainty inour estimate of n ( ω ) as well as the neglecting of the γ bandin the current calculation. We note that σ H , and therefore theestimated Kerr angle θ K , does depend on the actual ratio be-tween | ∆ A u | and | ∆ A u | . However, the dependence is quiteweak as long as the two are comparable [37]. SPONTANEOUS CURRENT
A chiral p-wave state shall generate finite spontaneous sur-face current [39, 40]. Following the original proposal of chi-ral p-wave pairing in Sr RuO , attempts to detect signaturesof such a current have had little success, suggesting that thespontaneous current is either simply absent or too tiny to beresolved in actual measurements [23, 24].The A u + iA u state has just the right appeal for explain-ing this, which is already obvious in the single-orbital case.Here, the two spin species each generates an edge current,and these currents flow in opposite direction due to the oppo-site chirality. Moreover, the integrated current in fact vanishesin the BCS limit. However, thanks to the distinct decay lengthscales of the two contributions, the current distribution doesnot exactly cancel at each spatial point.Going beyond single-orbital model, the integrated currentmay not vanish due to SOC and other band structure ef-fects. However, the strong suppression persists, as is demon-strated in Fig. 3 for our two-orbital model. Neglecting the fastFriedel oscillations, the spontaneous current generally con-tains two canceling contributions characterized by different Figure 3. (color online) Distribution of zero-temperature sponta-neous edge current in our two-orbital model with a mixed helicalp-wave A u + iA u pairing on the quasi-1D d xz and d yz orbitals.The A u component is held fixed at ∆ A u = 0 . t . Inset: a com-parison of the spontaneous current in the mixed helical and chiralp-wave phases with similar gap amplitudes. These calculations fol-low those in Refs. [42, 43], and lattice constant is defined to be unity. decay length scales. For the calculations shown in the fig-ure, the net current in the mixed helical state is typically morethan twenty times smaller than that in a chiral p-wave statewith similar gap magnitudes (See inset of Fig. 3).It should be stressed that such suppression is inherent, as itis achieved without appealing to surface disorder, anisotropicband and gap structure — factors typically having the poten-tial to further reduce the spontaneous current, as is known forthe case of chiral p-wave [41–46]. In fact, this behavior isalso expected from the Ginzburg-Landau theory, at the low-est order of which the spontaneous current is related to cross-gradient bilinears of the involving order parameters, such as ∂ x ∆ ∗ A u ∂ y ∆ A u [45, 47]. In the present case, this lowestorder contribution vanishes by symmetry, and higher orderterms such as ∂ x ∆ ∗ A u ∂ y ∆ A u are needed to account for thesmall local current distribution. By the same token, similarphysics occurs for the chiral d x − y + id xy pairing [48]. Inpassing, we note that for the B g + iA g state [49], i.e. the d x − y + ig xy ( x − y ) pairing, the spontaneous current is ingeneral finite at the (100) surfaces and vanishes at the (110)surface, analogous to the situation in s + id xy pairing [40].Finally, the suppression in the A u + iA u phase is also in-dependent of the edge orientation, and the same phenomenol-ogy takes place at the domain walls separating regions of time-reversed pairings. SPIN SUSCEPTIBILITY AND KNIGHT SHIFT
Recent revised NMR measurements under external in-planemagnetic field reveal a substantial Knight shift drop below T c [19, 20]. This implies a drop of spin susceptibility, χ spin ,which was subsequently confirmed by a polarized neutronscattering study [50]. A latter NMR study further places an Figure 4. (color online) Temperature dependence of the spin suscep-tibility in the A u + iA u (solid curves) and B g (dashed curves)superconducting states, in the presence of a small Zeeman field inthe x -direction. Here, η denotes the strength of SOC. upper bound of the normal state susceptibility at the lowtemperatures [25]. These reports are most straightforwardlyexplained in terms of spin-singlet pairing. However, as theobserved Knight shift contains contributions from both spinand orbital degrees of freedom, the exact fraction of the spinsusceptibility drop in the zero field limit may deserve a morecareful inquiry [10]. As we argue below, our proposed statesmay still have the potential to reconcile with the susceptibilitydrop.Owing to the in-plane d -vector orientation, the mixed heli-cal states naturally exhibit a reduced χ spin across T c under thein-plane field. In fact, in the absence of SOC, χ spin ( T = 0) equals to half of that in the normal state. This is demonstratedin Fig. 4, which shows the temperature-dependent susceptibil-ity in the presence of a weak Zeeman field directed along the x -direction. For comparison, we also show results for a spin-singlet d-wave state (a pairing in the B g irrep). In the calcu-lations, we have assumed that the two helical pairings onset atthe same critical temperature, and that their temperature de-pendence follows the standard BCS mean-field behavior.The SOC may facilitate extra spin-flipping processes thatcould have been otherwise suppressed by Cooper pairing,thereby alleviating the drop of χ spin . This is exemplified bythe noticeable increase of χ spin ( T = 0) as SOC becomesstronger (Fig. 4). Notably, even a pure spin-singlet pairingmay develop a residual χ spin ( T = 0) [29, 31]. For the two-orbital model in question, this can be explained in the fol-lowing terms. Here, SOC takes the form of η ˆ L z ˆ s z where ˆ L z operates on the orbital degrees of freedom and ˆ s z is thePauli matrix. The SOC term flips the in-plane spin compo-nents but not the one along z . As a consequence, upon turn-ing on a finite η , the z -component of the spin susceptibility χ zz spin remains fully suppressed at T = 0 (not shown in Fig. 4),whereas the x -component, χ xx spin , acquires a residual value. Weexpect such an enhancement of χ spin ( T = 0) due to SOC topersist in a more general multi-orbital model of Sr RuO [51–53]. Finally, it is worth remarking that, the natural mixing of (real)-spin triplet and singlet pairings in the presence ofSOC [51, 54], which is not considered here, may also intro-duce corrections to the spin susceptibility.Note that our calculations were done in the non-interactinglimit. Correlations renormalize the normal and supercon-ducting state χ spin in different manners, reducing the ra-tio χ spin ( T = 0) /χ spin ( T = T c ) from its non-interactingvalue [55]. Based on an estimate of the Wilson ratio of2 [20, 56] and using the data for η = 0 . t in Fig. 4, we ob-tain χ spin ( T = 0) /χ spin ( T = T c ) ≈ for the A u + iA u state. This is a substantial overall suppression, although, atface value, still noticeably higher than the upper bound of suggested by the most recent NMR measurement [25].However, our crude approximation could be insufficient for aquantitative comparison, and a calculation based on a morerealistic correlated multiorbital model may be needed. Wealso take note of the potential complications in disentanglingthe orbital and spin contributions to the Knight shift [10, 57],which could affect the experimental interpretation. STRAIN AND ULTRASOUND
Uniaxial strain and ultrasound measurements reveal crucialinformation about the symmetry of the superconducting or-der [15, 16, 58]. Most informative among the existing experi-ments are the ones that probe the coupling between the orderparameters and the B g ( (cid:15) x − y ) and B g ( (cid:15) xy ) strains. Forour mixed helical state, the leading order coupling has the fol-lowing form in the Ginzburg-Landau free energy, f coupling = X i (cid:15) i ( α i | ∆ A u | + β i | ∆ A u | ) (6)where i = x − y , xy for the B g and B g strains, respec-tively, and α i ( β i ) are coupling constants. No other order pa-rameter bilinears is present. This free energy has two implica-tions. On the one hand, the mean-field critical temperatures ofthe two components shall both follow a quadratic dependenceon the strain. This seems to agree with the observed behaviorin several uniaxial strain measurements [59, 60]. We do notethat there is no solid evidence of the expected split transitionsin thermodynamic measurements [61, 62], although this couldbe explained away by arguments unrelated to symmetry [63].On the other hand, the absence of a linear coupling to (cid:15) i inEq. (6) disallows any discontinuity in the shear elastic moduli ( c − c ) / or c . This stands in contrast with recent re-ports, where a jump in the latter has been observed [15, 16].Interestingly, a linear coupling to the B g strain (associatedwith c ) is possible for mixed A u and B u , as well as mixed A u and B u states [32]. This suggests a way to reconcile ourproposed order with the ultrasound experiments: the breakingof crystalline rotational symmetry around lattice dislocationscan allow the B u and/or B u order parameters to locally con-dense against the backdrop of a bulk A u + iA u order, leadingto a local mixture of A u with B u (and/or A u with B u ) thatcouples linearly to the B g strain. CONCLUDING REMARKS
Under the assumption that the observed polar Kerr effect isa genuine response of superconducting Sr RuO in the cleanlimit, we proposed that the mixed helical p-wave A u + iA u and B u + iB u states represent faithful alternative candidatesto chiral superconducting states. We also discussed how thesestates may be compatible with several other key measure-ments. In addition, as shown in a recent study [64], the helicalstate may also explain the Pauli limiting behavior of the in-plane H c and the first order phase transition observed in the H − T phase diagram at low T and at H near H c [65, 67].The most serious challenge for our proposal is probably thesubstantial Knight shift drop observed. Although we haveargued that the latest NMR measurement [25] does not nec-essarily rule out our mixed helical states, much further workis needed to fully resolve this issue. Also, a full reconcili-ation with other important observations may require furtherexamination. For example, the presence of gapless excita-tions [66, 68–71] would place strong constraint on the detailedforms of the p-wave pairings. To this end, we take note of themultitude of possibilities made available by the multiorbitalnature of this material [51–53, 72–80].The mixed helical states we discuss are also expected to ex-hibit a finite spin polarization in the bulk, resulting a sponta-neous bulk magnetization field that may be probed by SQUIDtype measurements. However, the field scales as (∆ /E F ) ,which may be below the resolution of the existing SQUIDtechniques [23, 24, 81].As a final remark, the B g + iB g state (chiral d x − y + id xy pairing) also carries the essential features needed to explainthe various observations discussed in this work. Alternatively,in a spirit similar to that proposed in Ref. [82], one can con-sider that one component of the B g + iB g state is favored inthe clean limit, while the other component condenses aroundlattice dislocations. Note that this scenario would naturallyconform with the reported nodal line behavior. ACKNOWLEDGMENTS
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CALCULATION OF THE ANOMALOUS HALL CONDUCTIVITY
In the absence of SOC, the two spin blocks decouple. We can calculate σ ↑ H and σ ↓ H separately. Spin up block
The model we consider consists of the two quasi-1d orbitals, d xz and d yz , of Sr RuO . Written in the basis { c d xz ( k ) , c d yz ( k ) } the spin up block of the normal state Hamiltonian is ˆ H N ( k ) ≡ (cid:18) ξ xz ( k ) λ ( k ) λ ( k ) ξ yz ( k ) (cid:19) = (cid:18) − t cos k x − t cos k y − µ t sin k x sin k y t sin k x sin k y − t cos k y − t cos k x − µ (cid:19) . (1)The corresponding BdG Hamiltonian is ˆ H ↑ BdG ( k ) = (cid:18) ˆ H N ( k ) ˆ∆ ↑↑ ( k )[ ˆ∆ ↑↑ ] † ( k ) − [ ˆ H N ] T ( − k ) (cid:19) (2)with ˆ∆ ↑↑ ( k ) = − (∆ A u − ∆ A u ) (cid:18) sin k x − i sin k y (cid:19) , (3)which is effectively a (spinless) chiral p x − ip y order parameter. For simplicity we have considered only intra-orbital pairingwhile neglected any inter-orbital one. The pairing component between the d xz and d xz orbitals, ∆ ↑↑ , transforms like k x underthe spatial D group. Hence, ∆ ↑↑ ( k ) = (cid:2) d ( x ) k · σ iσ (cid:3) ↑↑ = − (∆ A u − ∆ A u ) sin k x , where d ( x ) k is the k x -like part of d k = ∆ A u (ˆ x sin k x + ˆ y sin k y ) + ∆ A u (ˆ x sin k y − ˆ y sin k x ) . Similarly, ∆ ↑↑ ( k ) = (cid:2) d ( y ) k · σ iσ (cid:3) ↑↑ . From H BdG we define theelectric current velocity operator ˆ v ( k ) = (cid:18) ∇ k ξ xz ( k ) ∇ k λ ( k ) ∇ k λ ( k ) ∇ k ξ yz ( k ) (cid:19) ⊗ × , (4)where × is the identity matrix in the Nambu particle-hole space.We calculate σ ↑ H using the standard one-loop Kubo formula σ ↑ H ( ω ) = iω π xy ( q = 0 , ω + i + ) − π yx ( q = 0 , ω + i + )2 , (5)where π xy ( q , iν m ) is the electric current-current density correlator, given by π xy (0 , iν m ) = e T X n X k Tr [ˆ v x ( k ) ˆ G ( k , iω n )ˆ v y ( k ) ˆ G ( k , iω n + iν m )] . (6)In this equation ˆ G ( k , iω n ) ≡ [ iω n − ˆ H ↑ BdG ( k )] − is the Green’s function. ω n = (2 n + 1) πT and ν m = 2 mπT are Fermionicand Bosonic Matsubara frequencies. The trace Tr is with respect to both the Nambu particle-hole and orbital spaces. e is thecharge of an electron. Using Eq. (2) in the expression of π xy (0 , iν m ) in Eq. (6), completing the Matsubara sum over ω n , andthen making the analytical continuation, iν m → ω + iδ , we obtain σ ↑ H ( ω ) e / ~ = X k F ↑ k E ↑ + ( k ) E ↑− ( k )[ E ↑ + ( k ) + E ↑− ( k )][( E ↑ + ( k ) + E ↑− ( k )) − ( ω + iδ ) ] , (7) a r X i v : . [ c ond - m a t . s up r- c on ] F e b where F ↑ k = 32 ( t − ˜ t )( t ) (∆ − ∆ ) (sin k x cos k y + sin k y cos k x ) sin k x sin k y . (8) E ↑± ( k ) are eigenvalues of ˆ H ↑ BdG ( k ) . E ↑± = s − α ± p α − β , (9a) α = − ( ξ xz + | ∆ ↑↑ | + ξ yz + | ∆ ↑↑ | + 2 λ ) , (9b) β = ( ξ xz + | ∆ ↑↑ | )( ξ yz + | ∆ ↑↑ | ) + λ + λ ([∆ ↑↑ ] ∗ ∆ ↑↑ + ∆ ↑↑ [∆ ↑↑ ] ∗ ) − λ ξ xz ξ yz (9c)For brevity we have suppressed the k dependence in these equations.Written out explicitly, the real and imaginary parts of σ ↑ H areRe [ σ ↑ H ]( ω ) = e ~ X k F ↑ k E ↑ + E ↑− ( E ↑ + + E ↑− )[( E ↑ + + E ↑− ) − ω ] , (10)Im [ σ ↑ H ]( ω ) = e ~ X k F ↑ k E ↑ + E ↑− π ω (cid:20) δ ( ω − ( E ↑ + + E ↑− )) − δ ( ω + ( E ↑ + + E ↑− )) (cid:21) . (11) Spin down block
The derivation of σ ↓ H is almost identical. The only difference is that in the definition of the BdG Hamiltonian, ˆ∆ ↑↑ is nowreplaced by ˆ∆ ↓↓ ( k ) = (∆ A u + ∆ A u ) (cid:18) sin k x i sin k y (cid:19) . (12)Correspondingly, σ ↓ H ( ω ) e / ~ = X k F ↓ k E ↓ + E ↓− ( E ↓ + + E ↓− )[( E ↓ + + E ↓− ) − ( ω + iδ ) ] (13)with F ↓ k = − t − ˜ t )( t ) (∆ + ∆ ) (sin k x cos k y + sin k y cos k x ) sin k x sin k y . (14)The expressions of E ↓± are also almost identical to those of E ↑± in Eq. (9), except that { ∆ ↑↑ , ∆ ↑↑ } are now replaced with { ∆ ↓↓ , ∆ ↓↓ } . Although the gap magnitudes can be quite different for spin up and down, min { E ↑ + ( k ) + E ↑− ( − k ) } , whichdetermines the onset frequency of Im [ σ ↑ H ]( ω ) (see Eq. (11)), is actually nearly the same as min { E ↓ + ( k ) + E ↓− ( − k ) } , becauseboth of them are governed by the orbital-hybridization parameter t [1] instead of the pairing gap magnitudes. This explainswhy, in Fig. 1, Im [ σ ↑ H ]( ω ) and Im [ σ ↓ H ]( ω ) become nonzero essentially at the same frequency.Comparing F ↓ k to F ↑ k in Eq. (8) we see that they carry opposite signs, leading to a partial cancellation between σ ↓ H and σ ↑ H (see Fig. 1). This sign difference comes from ˆ∆ ↓↓ having a chirality opposite to that of ˆ∆ ↑↑ . If we set either ∆ A u = 0 or ∆ A u = 0 , then F ↑ k = −F ↓ k , E ↑± ( k ) = E ↓± ( k ) , and σ ↓ H + σ ↑ H = 0 , as we expect. Also, under the sign flip, ∆ A u → − ∆ A u or ∆ A u → − ∆ A u , σ ↑ H ↔ − σ ↓ H so that σ H → − σ H . This can be seen from Fig. 2, where we plot σ H ( ω ) for a given frequency asa function of A u ∆ A u / ( | ∆ A u | + | ∆ A u | ) . σ H changes its sign because flipping the sign of either ∆ A u or ∆ A u changeswhether the spin-up p x − ip y or the spin-down p x + ip y pairing dominates. ESTIMATION OF THE KERR ANGLE
From the calculated σ H ( ω ) we can estimate the Kerr angle for ~ ω = 0 . eV using θ K ( ω ) = 4 πωd Im [ σ H ( ω ) n ( n −
1) ] , (15) - × - × - × - × - × - ℏω [ eV ] R e ( σ H ) [ e / ℏ ] - × - × - × - × - × - ℏω [ eV ] I m ( σ H ) [ e / ℏ ] FIG. 1. Spin ↑ (red dashed line), ↓ (dark green dotted line), and the total (blue solid line) contributions to the T = 0 Hall conductivity.Left: Re [ σ H ] ; Right: Im [ σ H ] . Band parameters used are [1]: t = µ = 0 . eV, ˜ t = 0 . t , and t = 0 . t . ∆ A u = 2∆ A u = ∆ max with ∆ max = 0 . meV [1]. - - - × - - × - × - × - Δ Δ /(| Δ |+| Δ |) σ H ( ℏ ω = . V ) [ e / ℏ ] ( X10 ) FIG. 2. Real (blue solid line) and imaginary (red dashed line) parts of the zero temperature σ H ( ω ) as a function of A u ∆ A u / ( | ∆ A u | + | ∆ A u | ) . ~ ω = 0 . eV. ∆ max = | ∆ A u | + | ∆ A u | = 0 . meV is kept a constant in this calculation. The plot shows that σ H is roughlylinear in A u ∆ A u / ( | ∆ A u | + | ∆ A u | ) . From the plot we see that σ H is odd in the relative sign between ∆ A u and ∆ A u , but even underthe interchange | ∆ A u | ↔ | ∆ A u | . where n = n ( ω ) is the complex index of refraction, given by n = p (cid:15) ab ( ω ) , (16) (cid:15) ab ( ω ) = (cid:15) ∞ + 4 πiω σ ( ω ) . (17) (cid:15) ab ( ω ) is the permeability tensor in the ab -plane. (cid:15) ∞ = 10 [1] is the background permeability. d = 6 . ˚ A is the inter-layerspacing along the c -axis. σ ( ω ) is the optical longitudinal conductivity. Following Ref. [1] we use a simple Drude model for σ ( ω ) σ ( ω ) = − ω pl πi ( ω + i Γ) , (18)where ω pl = 2 . eV is the the plasma frequency and Γ = 0 . eV is an elastic scattering rate. At ~ ω = 0 . eV, σ ( ω ) =0 .
33 + i . , (cid:15) ab = − .
52 + i . , and n = 1 .
53 + i . . Plugging this n value into Eq. (15) and using the σ H ( ~ ω = 0 . eV ) value from Fig. 2 we get θ K ≈ nrad. The whole frequency dependence of θ K is shown in Fig. 3. [1] E.Taylor, and C. Kallin, Phys. Rev. Lett. 108, 157001 (2012). - - - ℏω [ eV ] θ K [ n r ad ] FIG. 3. (color online) Kerr angle as a function of ~ ω . Band parameters and the values of { ∆ A u , ∆ A u }}