Theory of chiral p -wave superconductivity with near-nodes for Sr 2 RuO 4
Wan-Sheng Wang, Cong-Cong Zhang, Fu-Chun Zhang, Qiang-Hua Wang
TTheory of chiral p -wave superconductivity with near-nodes for Sr RuO Wan-Sheng Wang,
1, 2, ∗ Cong-Cong Zhang, Fu-Chun Zhang,
3, 4 and Qiang-Hua Wang
2, 4, † Department of Physics, Ningbo University, Ningbo 315211, China National Laboratory of Solid State Microstructures & School of Physics, Nanjing University, Nanjing, 210093, China Kavli Institute for Theoretical Sciences & CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
We use functional renormalization group method to study a three-orbital model forsuperconducting Sr RuO . Although the pairing symmetry is found to be chiral p -wave, the atomicspin-orbit coupling induces near-nodes for quasiparticle excitations. Our theory explains a majorexperimental puzzle between d -wave-like feature observed in thermal experiments and the chiral p -wave triplet pairing revealed in nuclear-magnetic-resonance and Kerr effect. PACS numbers: 74.20.-z, 71.27.+a, 74.20.Rp
Introduction : The layered perovskite ruthenateSr RuO is one of the rare candidate materials thatis expected to carry chiral p -wave pairing in thesuperconducting (SC) state. Nuclear magnetic resonance(NMR) [1–4] and spin polarized neutron scattering[5] measurements show absence of drop in the spinsusceptibility below the SC transition temperature T c , providing identification of spin-triplet pairing inSr RuO . Muon spin relaxation [6] and polar Kerr effect[7] experiments reveal that time reversal symmetry inSr RuO is spontaneously broken below T c , suggestingchiral p -wave triplet pairing. The d -vector of the tripletis proposed as ˆ z ( k x ± ik y ),[8, 9] which is analogous tothat in the superfluid He-A phase. [10] In this case,the SC state is likely fully gapped, since no symmetryforces the chiral p -wave gap function to vanish on thequasi two-dimensional Fermi surface (FS) of the layeredSr RuO .In experiments, however, low-energy quasi-particleexcitations deep in the SC state, characteristic of gapnodes on the FS (forming nodal lines along the directionperpendicular to the RuO plane), are observed inspecific heat, [11–13] superfluid density, [14] spin-latticerelaxation rate, [15] thermal conductivity [16–18] andultrasound attenuation [19] at low temperatures. Toexplain the nodal-like behavior, a simple scenario is toassume d -wave pairing symmetry, so that the gap nodesare symmetry protected. This scenario is, however,inconsistent with the compelling signatures of chiral p -wave triplet mentioned above. An alternative scenariois the chiral p -wave gap function may have deep minimaor accidental nodes.[20–24] The linear specific heat andthermal conductivity below T c / min should be much smaller than the gapmaximum ∆ max .[21, 22] The recent thermal conductivitymeasurement [18] sets an upper bound ∆ min / ∆ max ≤ / d -wave pairing,or d -wave-like f -wave pairing in the form of ( k x + ik y ) g ( k ), where g ( k ) ∼ k x k y or k x − k y .[25–27]Sr RuO has three energy bands ( α , β and γ , derived from the d xz,yz,xy orbitals) crossed by the Fermi level,with the γ Fermi pocket closer to the van Hove singularity(vHS) on the zone boundary. The singular-modefunctional renormalization group (SMFRG) study of thethree-orbital model without spin-orbit coupling (SOC)[24] showed that the gap function on the γ pocket islargest and strongly anisotropic, with ∆ min / ∆ max ∼ /
10. However, such a gap structure is not yetenough to explain the linear specific heat and thermalconductivity at the measured low temperatures. Modelswith SOC were previously studied by using weak couplingRG and random phase approximation.[28, 29] But toour knowledge, close and systematic comparisons toexperiments have not been reported.The outstanding puzzle of the chiral p -wave pairingrevealed in NMR and Kerr effect and the d -wave-likebehavior indicated in thermal experiments motivatesus to perform more careful microscopic investigations.We consider a comprehensive model including all of thethree orbitals and the atomic SOC.[30–33] We adoptthe band structure (with the effect of SOC) that bestfits the angular-resolved photo-emission spectroscopymeasurement.[34] We apply the spin-resolved versionof SMFRG [35–38] and treat all possible orderingtendencies on equal footing.Our main results are summarized in Figs.4 and 5. Wefind that chiral p -wave pairing is dominant and can berelated to the small-momentum spin fluctuations derivedfrom the d xy orbital, similarly to the case in Ref.[24].However, SOC induces near-nodes on the γ pocket,with ∆ min / ∆ max < / α and β pockets. Thecalculated specific heat, superfluid density, Knight shift,spin-lattice relaxation rate and thermal conductivity arein excellent agreement with experimental data, superiorto the d -wave fits. Our theory reconciles the d -wave-likefeature in thermal measurement and chiral p -wave spintriplet pairing in NMR and Kerr effect in Sr RuO . a r X i v : . [ c ond - m a t . s up r- c on ] J a n FIG. 1: (a) Band dispersion along high-symmetry cuts. (b)Fermi surface (lines) and the spectral weight of the d xy -orbital(color scale) thereon. Model and method : We now specify the modelHamiltonian H = H + H I for Sr RuO . The free partcan be written as H = (cid:88) k ψ † k h k ψ k , h k = (cid:15) k σ − λ L · (cid:126)σ/ . (1)Here ψ k = ( c k ↑ , c k ↑ , c k ↑ , c k ↓ , c k ↓ , c k ↓ ) T is thefermion spinor, with c k as annihilating an electron ofmomentum k and spin s ∈ ( ↑ , ↓ ) on orbital a ∈ (1 , , ↔ ( d xz , d yz , d xy ). In the single-particle Hamiltonian h k , (cid:15) k is a matrix in the orbital basis, L is the orbital angularmomentum, and (cid:126)σ/ λ = 0 .
032 eV,[34] and the other detailsfor h k can be found in Refs.[34, 39] Fig.1(a) shows theband dispersion calculated with H along high symmetrycuts. By inversion and time-reversal symmetries, eachband is doubly degenerate in pseudo-spin.[39] Fig.1(b)shows the Fermi surface (FS). Note the d xy -content ofthe Bloch state is dominant on the γ pocket, but vanishesidentically along G-M.The interacting part of the Hamiltonian H is given by,in real space, H I = U (cid:88) ia n ia ↑ n ia ↓ + J (cid:88) i,a>b,ss (cid:48) c † ias c ibs c † ibs (cid:48) c ias (cid:48) + U (cid:48) (cid:88) i,a>b n ia n ib + J (cid:48) (cid:88) i,a (cid:54) = b c † ia ↑ c † ia ↓ c ib ↓ c ib ↑ , (2)where i denotes the lattice site, n ia = (cid:80) s c † ias c ias , U is the intra-orbital repulsion, U (cid:48) is the inter-orbitalrepulsion, J is Hund’s rule coupling, and J (cid:48) is thepair hopping term. The interactions can lead tocompeting collective fluctuations in particle-hole (PH)and particle-particle (PP) channels, which we handleby SMFRG. Following the general idea of FRG,[47] we obtain the one-particle-irreducible 4-pointinteraction vertices Γ (where numerical index labelssingle-particle state) for quasi-particles above a runninginfrared energy cut off Λ (which we take as the lowerlimit of the continuous Matsubara frequency). Startingfrom Λ = ∞ where Γ is specified by the bare parametersin H I , the contribution to the flow (toward decreasing FIG. 2: One-loop diagrams contributing to ∂ Γ /∂ Λ,quadratic in Γ itself (wavy lines, fully antisymmetrized withrespect to incoming or outgoing fermions, labelled by thenumerical indices). The color of the wavy line signifies thescattering of fermion bilinears in the pairing (blue), crossing(red) and direct (green) channels.
Λ) of the vertex, ∂ Γ /∂ Λ, is illustrated in Fig.2.At each stage of the flow, we decompose Γ in termsof eigen scattering modes (separately) in the PP andPH channels to find the negative leading eigenvalue(NLE), the divergence of which signals an emergingorder at the associated scattering momentum, withthe internal microscopic structure described by theeigenfunction. The technical details can be foundelsewhere,[24, 35–38, 48–52] and also in Ref.[39].
Discussions : We consider the bare interactionparameters (
U, U (cid:48) , J, J (cid:48) ) = (0 . , . , . , .
04) eV. Theresults are qualitatively robust against fine tuning ofinteractions and SOC around the present setting.[39]Fig.3(a) shows the flow of NLE S PH (among all momenta)in the PH channel. The corresponding momentumchanges from Q ∼ (0 . , . π at high energy scaleto Q ∼ (0 . , . π in the intermediate stage. Wechecked that the eigenfunction describes site-local spin.The d xz,yz ( d xy ) orbitals dominate before (after) thelevel crossing. The inset shows the NLE S PH ( q ) asa function of momentum q at the final stage of theFRG flow. We see a strong peak at Q and alsoa secondary peak at Q . These peaks are consistentwith the spin-fluctuations observed in neutron scatteringexperiments.[53] Our results provide clear origins of suchpeaks: spin fluctuations at Q ( Q ) arise mainly fromthe d xz,yz ( d xy ) orbital, similarly to the case withoutSOC.[24] At low energy scales, the PH channel saturatesdue to the decreasing phase space for low-energy PHexcitations.Fig.3(b) shows ten NLE’s in the PP channel (at zeromomentum). They are induced at intermediate scales,where the PH channel is enhanced, a manifestation of theKhon-Luttinger mechanism,[46] namely, the interactionin the PH channel has an overlap in the PP channel.Eventually, a particular mode (red thick line) diverges.We find it describes p x,y -wave pairing (to be detailed FIG. 3: (a) Flow of negative leading eigenvalue (among allmomenta), S PH in the PH channel, shown as 1 /S PH for clarity.The inset shows − S PH ( q ) in the momentum space at thedivergence scale Λ = Λ c . (b) Flow of NLE’s S PP ( q = 0).The thick line denotes the two eventually diverging p − wavepairing modes. Arrows indicate level crossing for Q /π in thePH channel (a) and the pairing symmetries (b). below), and is twofold degenerate by C v symmetry. Thedetails of the pairing function (the eigenfunction of theNLE scattering mode in the PP channel) are presentedin Ref.[39] Here we show the projection of the p x + ip y pairing function (favored in the SC state) in the bandbasis in Fig.4. There are several remarkable features: (i)In Fig.4(a), the phase of the gap function changes veryrapidly across G-X. This follows from anti-phase pairingbetween d xy -electrons on first- and second-neighborbonds.[39] (ii) Figure 4(b) shows a gap minimum at θ = 0 on the γ pocket, with ∆ min / ∆ max < / θ = 0) furtherby more than one order of magnitude, in comparisonto the gap (dashed line) when SOC is artificially set tozero.[54] (iii) On the γ pocket, the gap is also small at θ = 45 o (or along G-M), which would be close to thegap maximum without SOC. This feature is related tothe fact that the d xy -weight is missing on the Fermipocket along G-M (see Fig.1), whereas the dominantpairing component involves d xy -orbital.[39] (iv) SOCalso induces sizable and anisotropic gaps on the α and β pockets, significantly larger than that without SOC.[24]We calculate various properties of the SC state usingthe FRG-derived mean field theory,[39] and compare tothe experimental data. No other tuning parameters areinvoked regarding the gap structure.[55] The results arepresented in Fig.5. In the experimental regime, our gapstructure behaves effectively nodal, and could in fact fitthe data better than that in the d -wave case suggestedin Ref.[18]. The details are as follows. FIG. 4: (a) FRG-derived p x + ip y -wave gap function on the FS(thin black lines). The arrow represents (Re ∆ k n , Im ∆ k n )for n ∈ ( α, β, γ ). (b) The solid lines show the gap amplitude(up to a global scale) on the FS versus the Fermi angle θ in aquadrant of the respective pocket. The dashed line shows thegap on the γ pocket if SOC is switched off artificially, showingthe effect of SOC in generating deeper near-node along G-X( θ = 0) and local minimum along G-M ( θ = 45 o ). In Fig.5(a), we show the specific heat in our chiral p -wave case (solid line), which is in excellent agreementwith the experimental data (symbols) extracted fromRef.[11], both in the quasi-linear behavior below T c / T c . In comparison, the d -wave fit (dashedline) is much poorer in both aspects.In Fig.5(b) we show the superfluid density ρ . Theexperimental data (symbols) are extracted from Ref.[14]where T c = 1 . ζ from nonmagnetic impurities in the experimentalsituation as,[56, 57]ln( T c /T c ) = Ψ(1 / ζ/ πT c ) − Ψ(1 / , (3)where Ψ( x ) is the digamma function, T c = 1 . K is assumed to be the transition temperature in thedisorder-free material. We get ζ/T c ∼ . T c = 1 .
39K according to Eq.3. Using this value of ζ , the resultfor the chiral p -wave (green line) deviates from the data(symbols) in view of the curvature in the intermediatetemperature window. However, if we assume ζ/T c = 0 . T c might be even higher than 1.5K. In comparison,the d -wave fits (dashed lines) for both scattering ratesdeviate from the data.The spin-lattice relaxation rate 1 /T is shown inFig.5(c). The theoretical result in our chiral p -wave case(solid line) is in good agreement with the experimentaldata (symbols) extracted from Ref.[15] (where T c =1 . K corresponds to ζ/T c = 0 .
02 via Eq.3). Notethe approximate power-law behavior 1 /T ∝ T in theintermediate temperature regime. The d -wave fit (dashedline) show similar but slightly poorer agreement. TheKnight shift K µµ depends on the probed spin direction µ , see Fig.5(d). K xx,yy barely changes, while K zz issuppressed below T c . This is because our pairing function FIG. 5: The calculated physical properties in the SC state(lines), in comparison with experimental data (symbols). Thesolid lines are for our chiral p -wave, while the dashed linesare d -wave fits. (a) The electronic specific heat C versus thetemperature T . Here γ n is the (constant) value of C/T inthe normal state. The symbols are extracted from Ref.[11],where T c = 1 . K . (b) Superfluid density ρ versus T , withsymbols from Ref.[14]. (c) Spin-lattice relaxation rate 1 /T versus T , normalized with respect to the value at T c . Thesymbols are from Ref.[15], where T c = 1 . K . (d) Thedirection-resolved Knight-shift K versus T . (e) Thermalconductivity κ versus T . The symbols are from Ref.[18]. Herewe use ζ/T c = 0 .
26 according to the experimental T c = 1 . κ/T as a function of impurityscattering rate ζ . The temperature is fixed at T = T c / (cid:15) n , the value of κ/T at T = T , ζ = 0 . T c , and zero gap. is dominated by the triplet component with its d -vectoralong z ,[39] so that the spin of the Cooper pair lies inthe plane and can response, in the linear limit, to weakin-plane (out-of-plane) field without (by) pair breaking.In experiment, K zz is also unchanged below T c , andthis is explained by the fact that the experimental fieldis large enough to rotate the d -vector, given the smallenergy gap. [2, 58]Figure 5(e) shows the calculated κ/T (lines) versus T with ζ/T c = 0 .
26, along with the experimental data(symbols) with T c = 1 . p -waveresult (solid line) agrees to the data much better thanthe d -wave case (dashed line), in view of the curvaturein the intermediate temperature window. Figure 5(f) shows the calculated κ/T versus ζ (lines) at the fixedlow temperature T = T = T c /
30, compared to theexperimental data (symbols) from Refs.[16–18]. Wesee our chiral p -wave case (solid line) fits the datavery well, including the universal behavior [59] at ζ/T = 30 ζ/T c (cid:29)
1, and the mild decrease near andbelow ζ/T c = 0 .
4. In contrast, in the d -wave case (dashedline) κ/T increases monotonically with decreasing ζ ,although it also shows universal behavior on the large- ζ side. (Note the eventual rise as ζ/T → d -wave case,[39, 59] but in both cases can be explainedby a Boltzman equation for well-defined quasiparticles,which predicts κ/T ∝ /ζ . On the other hand, wehave normalized the numerical κ/T by (cid:15) n , the valueof κ/T with T = T , ζ = 0 . T c , and zero gap. Thisleaves the relative size of κ/T in the p - and d -wavecases unambiguous.) Therefore, the experimental data,rather than implying d -wave pairing, actually supportsa gap structure with various gap minima on the threeFermi pockets, as in our chiral p -wave case. This issupported by further discussions in Ref.[39] Of course,if the probing temperature T is reduced further, sothat T (cid:28) ∆ min , κ/T is eventually suppressed.[39] Atthis stage the d -wave and chiral p -wave behave mostdifferently. Measurement at such low temperatures isimportant to close the issue, but might be a challenge inexperiment. Summary and remarks : We studied the super-conductivity of Sr RuO by the state-of-art SMFRG.We find that chiral p -wave pairing is dominant, butSOC induces deep near-nodes on the γ pocket and alsosizable and anisotropic gaps on the α and β pockets.The microscopic theory is in excellent agreement withexperiments, resolving the outstanding puzzle betweenthe d -wave-like feature in thermal measurements and thechiral p -wave superconductivity revealed in NMR andKerr effect experiments.Remarkably, the simultaneous presence of deepestnear-nodes along G-X and less deep ones along G-M(both on the γ pocket in our case) is exactly thegap structure speculated to explain the systematicangle-dependent specific heat under inplane as well asconical magnetic fields in Ref.[60], where the near-nodesalong G-M were assumed (but do not have) to be onthe α and β pockets. The near-nodes may also bean important factor to reduce the spontaneous edgecurrent (not detected so far[61]) at finite temperaturesand under impurity scattering.[62–64] We leave these asfuture topics.The project was supported by the National KeyResearch and Development Program of China (underGrant No. 2016YFA0300401), the National BasicResearch Program of China by MOST (under GrantNo. 2014CB921203), and the National Natural ScienceFoundation of China (under Grant Nos.11604168,11574134, 11674278, and 11404383). FCZ alsoacknowledges the support by the Strategic PriorityResearch Program of the Chinese Academy ofSciences (under Grant No. XDB28000000). WSWalso acknowledges the support by K. C. Wong MagnaFundation of Ningbo University. ∗ Electronic address: [email protected] † Electronic address: [email protected] [1] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q.Mao, Y. Mori, and Y. Maeno, Nature(London) , 658(1998).[2] H. Murakawa, K. Ishida, K. Kitagawa, Z.Q. 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