Maximization of T c via Conspired Even-Parity Spin and Charge Collective Excitations in Strained Sr 2 RuO 4

Transcription

1 arxiv: v1 [cond-mat.str-el] 13 Nov 18 Maximization of T c via Conspired Even-Parity Spin and Charge Collective Excitations in Strained Sr RuO 4 Swagata Acharya 1, Dimitar Pashov 1, Cédric Weber 1, Hyowon Park, Lorenzo Sponza 3, Mark van Schilfgaarde 1 1 King s College London, The Strand, WCR LS London, UK, Department of Physics, University of Illinois at Chicago, Chicago, Illinois 667, USA, 3 LEM UMR 14, ONERA-CNRS, F-93, Châtillon, France swagata.acharya@kcl.ac.uk. Unconventional superconductivity has been intensively studied for decades, yet the origin and nature of the pairing is often obscure. This is particularly the case in Sr RuO 4 : there is no consensus on the driving forces for the superconducting gap, its symmetries and dimensionality after a quarter of a century of detailed study. Multiple low-energy scales in spin, charge and orbital degrees of freedom are present, which make it difficult to ascertain the validity of model approaches which make assumptions that exclude or single out particular mechanisms. Here we develop an alternative approach, a detailed ab initio theory that can adequately treat both local and non-local spin and charge correlations, and multi-particle vertex functions in all relevant degrees of freedom, that allow calculation of spin, charge, and pairing susceptibilities. Recent advances that combine the quasiparticle self-consistent GW approximation, a perturbation theory that accurately captures long range charge cor- 1

2 relations with dynamical mean field theory (DMFT), which includes local spin fluctuations, make this possible. We find that Sr RuO 4 has a superconducting gap structure with nodal lines along all three crystalline directions. Spin fluctuations, dominated by the highly incommensurate vector, tries to stabilize a singlet pairing state but the Ru-t g manifold drive strong inter- and intraorbital charge fluctuations in triplet channel. When subject to stress, the otherwise competing spin and charge channels are found to co-operate and drive an even-parity pairing, which causes T c to increase. This occurs up to a critical stress, beyond which spin fluctuations become less coherent in singlet channel and T c begins to fall. Superconductivity (1) is an emergent quantum property of a material where electron participate in dissipation-less charge transport. While electrons repel each other via the Coulomb force, quantum theory provides additional interactions that in special circumstances at low temperature can surmount the repulsion to bind two of them together to form Cooper pairs (). When the pairing is mediated by coupling to nuclear motion (phonons) it is called conventional superconductivity and it is well understood in terms of the celebrated BCS theory (3). Unconventional superconductivity can originate from the interplay of degrees of freedom other than phonons. It usually involves spin fluctuations but a general understanding of its origin is lacking. Here we focus on Sr RuO 4, an unconventional superconductor which is highly sensitive to disorder (4). Bulk single crystals show superconductivity below 1.5 K (5), though the superconducting pairing and the associated order parameter lack comprehensive understanding (6), 4 years after it was observed. Sr RuO 4 is of great interest because the superconductivity may have spin-triplet symmetry, which raises the possibility that it can sustain Majorana states conducive for topological quantum computing (7). The superconducting transition temperature, T c, has been observed to increase to 3 K in

3 eutectic crystals of Sr RuO 4, in the vicinity of Ru inclusions (8, 9, 1). While enhancement of T c was traditionally observed with a lowered volume fraction, a recent series of experiments on bulk single crystals of Sr RuO 4, subject to uniaxial strain show a similar increment (11,1): increases to 3.4 K were observed for tensile strain in the [1] direction, which we denote as ɛ x. These apparently dissimilar studies hint towards a more common underlying mechanism for enhancement of T c : Ru inclusions induce local stresses which include uniaxial strain. In the tensile experiments T c can be controlled by varying ɛ x. It reaches a maximum value at ɛ x =.6% (1), beyond which it falls rapidly. These observations challenge the established belief that Sr RuO 4 is a spin-triplet (oddparity) superconductor. Since tetragonal symmetry is broken, it is no longer possible to find an order parameter with two degenerate components such as p x +ip y or d xz +id yz. Hence this raises a further possibility that even if the unstrained compound is odd-parity superconductor, strain could induce strong even-parity components. Subsequent DFT studies (1), weak coupling studies, single-band minimal Hamiltonian studies (13, 14) identified a change in Fermi surface topology with ɛ x, with an attendant Van Hove singularity (1) that approaches the Fermi energy E F. It was proposed (15) that T c is enhanced by an increase in free carriers as the singularity crosses E F, then falls off after the peak passes through it. Such a picture identifies an important property resulting from strain, but it is not sufficient to explain the enhancement of T c. Sr RuO 4 is a classic instance of multi-orbital electron itinerary (16, 17, 18, 19). Electron correlations originate from competition between intra-orbital Hubbard correlations and sizeable Hund s coupling (, 16, 1). In a multi-orbital scenario with large Hund s coupling, the splittings that emerge after the tetragonal xy degeneracy is lifted, are likely to play a pivotal role in affecting T c. They are left out of RG based studies based on a minimal, single-band Hamiltonian. Additionally, spin-flip scattering and the pair-hopping terms absent in a minimal single-band Hamiltonian are very important for Sr RuO 4. Ab initio studies based on DFT pro- 3

4 vide some basic insight as noted above. But correlations and the attendant self-energy Σ(k, ω) are highly non-local in space, momentum and time; these significantly modify DFT eigenvalues and eigenfunctions. DMFT added to DFT restores the local part of correlations, notably spin fluctuations on Ru, but it cannot correct for DFT s failure to adequately capture the underlying electronic structure. Its tendency to misalign O-p and Ru-d, and overestimate interatomic couplings, change both the dispersions and the orbital character of crystal eigenstates in important ways. In short, the complexity of Sr RuO 4 warrants a theoretical technique that incorporates all of the important correlations in a unified ab initio framework. We implement a recently developed three-tier technique starting with the quasi-particle self consistent GW (QSGW) approximation to build our reference hamiltonian, augment it with dynamical mean field theory (DMFT) to add spin fluctuations left out of QSGW, and also generate the vertices entering into spin, charge, and pairing susceptibilities. Finally we solve multi-orbital Bethe-Salpeter equations to calculate these properties in both strained and unstrained single crystals. We identify what leads to the non-monotonic dependence of T c on strain and also gain insights into the upper bound for T c in Sr RuO 4. We find a one-to-one correspondence between T c and the coherence and intensity of the spin and charge susceptibilities under application of strain. Finally, we establish connections between spin, charge fluctuations and superconducting pairing symmetries in Sr RuO 4, and dimensionality of the fluctuations associated with these degrees of freedom. The local density of states (DOS) from QSGW shows the presence of a Van-Hove singularity slightly above the Fermi level in the unstrained case (Fig. 1). Strain splits the single peak, the splitting increasing linearly with ɛ x. The smaller, lower peak crosses E F at ɛ x =.6%, which we denote as ɛ x. We note in passing that if the larger Van-Hove feature could be tuned to cross E F, it may be possible to induce a different, and possibly stronger superconducting transition. Strain lifts the tetragonal symmetry, splitting the degeneracy of the Ru (4d xz, 4d yz ) and O (p x, p y ) 4

5 pairs. We find that the states split beyond the typical exchange scale of 4 mev (Fig. 8) when ɛ x =.6%. Fermi surfaces in the basal plane are shown in Fig.. The critical change in topology on the line connecting (,) and (,π) (points Γ and M) occurs at ɛ x=.6%, in excellent agreement with strain where T c maximizes and where one Van-Hove singularity crosses E F (Fig. 1), as was noted in a a prior DFT study; though in DFT ɛ x is much larger (see Fig. ). The green wobbly contour is clearly observed in high-resolution angle-resolved photo-emission spectroscopy (). That QSGW simultaneously yields ɛ x near the observed maximum T c, and can reproduce fine details of the ARPES Fermi surface, is a reflection its superior ability to generate good effective noninteracting hamiltonians. Coulomb interactions are long-range; they are treated by QSGW perturbatively in a selfconsistent manner through a dynamic and momentum dependent self-energy, Σ(k, ω). We analyse Σ(k, ω) computed within QSGW to extract the momentum-dependent quasi-particle renormalization factor Z k = (1 Σ(k, ω)/ ω) 1. Fig. 3 shows how the non-local Σ(k, ω) renormalizes bands differently on the Γ-M line. Notably the renormalization is stronger away from the Fermi energy (ω=). The Ru 1 band which has d xy character on this line and which participates in the Fermi surface reconstruction, renormalizes most strongly, Z k varying by more than 4% on the line with Ru-d xz,yz varying by 3%. With strain the k dependent renomalization evolves in a non-trivial fashion, with Z k,dxy varying by 3% at critical strain of.6% and nearly 5% beyond that. We next consider two-particle responses of the system, and beginning with the dynamical spin susceptibility χ s (q, ω) on the Γ-X line, initially for the bulk compound, and then how it is modified by strain. Fig. 4 shows momentum and energy resolved χ s (q, ω) for ɛ x =%,.6% and 1.% in the planes q z =, 1/4, and 1/ (in units of π/c). The unstrained compound shows a dominant peak at incommensurate (IC) q=(.3,.3, ) (3, 4, 5) with maximum near 5

6 ω=1 mev. However it disperses all the way up to 8 mev, which is in excellent agreement with experimental observations (6). We also find significant spin fluctuations at the ferromagnetic (FM) vector q = (,, ) (also seen in very recent neutron measurements (7)) and almost no intensity at the antiferromagnetic nesting vector (1/,1/,). The FM signal is important, because it arises with spin fluctuations having multi-orbital character and also because it has implications for superconductivity (7). We find that the intensity of χ s (q=) is 1/5 of the dominant IC peak when spin-orbit (SO) coupling is suppressed. But SO coupling lifts band degeneracies at high symmetry points, reducing this ratio to 1/8. Thus χ s seems to be dominated by fluctuations at the IC vector. Hence, it should favour pairing mainly in the singlet channel, unless there are other novel channels to provide extra glue for a triplet pairing. As strain is applied, χ s changes in significant ways. Imχ s (q, ω) becomes sharper and more coherent: the peak at χ s (q=(.3,.3, ), ω=1 mev) nearly doubles when ɛ x=.6%. For still larger ɛ x, the increasing coherence of Imχ s begins to reverse. At ɛ x =1.%, the IC peak (.3,.3, q z ) survives but χ s becomes incoherent and diffused over a range q, with another peak near (.15,.15, ) becoming apparent. Note also that Imχ s also acquires a dependence on q z : At ɛ x =, χ s depends weakly on q z ; for ɛ x =.6% the q z dependence is significant and the dominant peak is most intense at 1/4. For ɛ x > ɛ x, Imχ s begins to acquire intensity at (1/, 1/, q z ), with strong dependence on q z. In short, for ɛ x > ɛ x, two prominent changes are observed: incommensurate but nearly ferromagnetic excitations at (.15,.15, ) and commensurate antiferromagnetic spin excitations at (1/, 1/, a/c). At what wave vector spin and charge fluctuations are strong is of central importance to the kind of superconducting pairing symmetry they can form. If superconductivity is driven by fluctuations near the ferromagnetic point (,,), the spin part of the Cooper pair is symmetric and the superconductivity should have triplet symmetry. If, on the other hand if the fluctuations (spin or charge) are more proximate to (1/, 1/, q z ), the symmetry is more likely to be singlet. We 6

7 compute the particle-particle (p-p) vertex functions from our ab initio approach, which enables us to determine what predominanates the driving forces for pairing. The p-p vertex functions are connected to particle-hole (p-h) vertex functions in the magnetic and charge channels (8). The particle-particle polarization bubble is dressed with the p-p vertex functions to get the susceptibilities in the particle-particle channel. Finally we solve the non-linear eigenvalue problem of the BCS-equation to get the pairing fields for all eigenvalues. This construction allows us to separate the contributions to the superconducting pairing into spin and charge contributions. Fig. 6 shows the pairing fields for leading five eigenvalues along the reciprocal lattice plane determined by vectors connecting (,,) (1,,) and (,,) (,1,). The first two eigenvalues are degenerate, and start to decay with the third eigenvalue. The fifth eigenvalue, approximately half of the leading one, makes the spectrum extremely dense. For the first five eigenvalues, the details of the gap structure vary significantly, however, the underlying qualitative features for the nodal and anti-nodal gap lines of the pairing fields remain invariant (Fig. 6). Most importantly the nodal features exist both in-plane and out-of-plane in the Cartesian representation (Fig. 6 shows the pairing field in crystal coordinates). Nodal gap minima, nodal points and nodal lines have been predicted from prior experiments and minimal Hamiltonian calculations in the unstrained compound (9, 3, 31). However, only recently has pure nodal line gap character been observed experimentally along the Cartesian z-direction (3). Owing to technical limitations the experiments by Hassinger et al. (3) cannot comment on the character of the nodal gap structure in basal plane, but they find strong evidence for nodal lines along q z, making a strong case for the H c anomaly in Sr RuO 4. In a very recent specific heat study, under angular variation of magnetic field at very low temperatures, Kittaka et al. (33) established the presence of horizontal line nodes in the gap structure. Our finding that nodal gap structure exists all three spatial directions are in excellent qualitative agreement with these recent experimental observations, (a quantitative comparison is out of scope in the absence of experimental data). Under 7

8 strain, the gap character changes non-trivially, however, the fundamental nodal anti-nodal character remains invariant for all strains. The evolution of the spin and charge susceptibilities are instructive to understand the changes in the gap symmetries under strain and their underlying even- or odd-parity characters. We find that the real part of the charge susceptibilities in the static limit χ c (q, ω ), has strong peaks both at the the nearly ferromagnetic (.,.,) vector and also at more commensurate higher wavelength quasi-anti-ferromagnetic vector ( 1, 1,) (Fig. 5). A possible route to superconductivity through singular charge fluctuation originating from one dimensional bands is discussed by Raghu et al. (34). While we find them to be significant, we also observe nearly comparable fully multi-orbital charge fluctuation, both intra and inter-orbital in nature, in all active bands (Fig 5). Inter-orbital charge fluctuations originating from the quasi-one dimensional Ru-d xz,yz and two-dimensional Ru-d xy are comparable to, or even larger than the intra-orbital charge fluctuations. Nearly uniform long-wavelength coherent charge fluctuations would support a triplet pairing channel through multi-orbital charge fluctuations. In the unstrained case, there is a significant peak in χ q at small q, near (1/5,1/5,), (and a significant component of χ s as well, as noted above). However, there is also a peak near (1/,1/,). Under strain, the latter peak becomes more coherent and larger, while the former decays. At the critical strain ɛ x=.6% only the peak near (1/,1/,) remains. Also χ s at (.3,.3,q z ) becomes increasingly coherent; the two combine to favour a singlet pairing environment. This is strikingly different from the unstrained scenario where both spin and charge fluctuations had favourable triplet components as well. For ɛ x > ɛ x, charge fluctuation peak at (π, π) diverges. Simultaneous shifts in spin fluctuation weight towards more commensurate lower energy large q suggests an emergent spin-charge separation which is not conducive for pairing. We draw the following conclusions. Strong spin fluctuations at the (.3,.3, q z ), combined with the smaller peak at the quasi-ferromagnetic triplet vector are not enough to drive a purely 8

9 spin- triplet superconductivity. We find that non-locality in charge correlations plays a central role in Sr RuO 4 : they provide additional glue through strong intra- and inter-orbital fluctuations. More specifically, the charge fluctuations are unusually large in the Ru-d xz,yz -d xy inter-orbital coupling channel through a highly incommensurate (.,., q z ) vector. Together, in the unstrained case, the dominant spin fluctuation peak at (.3,.3, q z ), supplies glue for even-parity spin pairing and coherent charge fluctuation at (.,., q z ) vector (and incoherent quasi-antiferromagnetic peak) provide glue in orbital-triplet pairing channel. We find a superconducting gap structure with nodal and anti-nodal lines in all spatial directions in such competing spincharge scenario. Under strain the peak in χ s (q=.3,.3, q z ) becomes more coherent up to a critical strain ɛ x. Simultaneously the peak in χ c (q=.,., q z ) gets fully suppressed while becoming more coherent at the quasi-anti-ferromagnetic vector (.5,.5, q z ). Together they suggest, spin and charge co-operate to sustain an even parity pairing channel which maximize T c at ɛ x. For ɛ x >ɛ x, the spin fluctuation weight drifts toward larger wavelength, more uniform quasi-ferromagnetic vectors and charge fluctuates more strongly at the quasi-anti-ferromagnetic vector. This emergent spin-charge separation, split by quasi-ferromagnetic spin fluctuation peak and quasi-anti-ferromagnetic charge fluctuation peak, is not conducive for sustaining the evenparity superconductivity and hence lowers and suppresses T c. Our observations suggest that the pathway to maximize superconductivity in Sr RuO 4 is to make spin and charge fluctuations co-operate in an even parity channel. Acknowledgements This work was supported by the Simons Many-Electron Collaboration, and EPSRC (grants EP/M11631/1 and EP/M1138/1). For computational resources, we were supported by the ARCHER UK National Supercomputing Service and the UK Materials and Molecular Modelling Hub for computational resources, (EPSRC grant EP/P194/1). 9

10 Method We use a recently developed quasi-particle self consistent GW + dynamical mean field theory (QSGW+DMFT) (35, 36) approach to address this problem, as implemented in the Questaal package (37). Paramagnetic DMFT is combined with nonmagnetic QSGW via local projectors of the Ru 4d states on the Ru augmentation spheres to form the correlated subspace. We carried out the QSGW calculations in the tetragonal and strained phases of Sr RuO 4 with space group 139/I4mmm. DMFT provides a non-perturbative treatment of the local spin and charge fluctuations. We use an exact hybridization expansion solver, namely the continuous time Monte Carlo (CTQMC) (38), to solve the Anderson impurity problem. The one-body part of QSGW is performed on a k-mesh and charge has been converged up to 1 6 accuracy, while the (relatively smooth) many-body static self-energy Σ (k) is constructed on a k-mesh from the dynamical GW Σ(k, ω). Σ (k) is iterated until convergence (RMS change in Σ <1 5 Ry). U = 4.5 ev and J = 1. ev (39) were used as correlation parameters for DMFT. The DMFT for the dynamical self energy is iterated, and converges in 1 iterations. Calculations for the single particle response functions are performed with 1 9 QMC steps per core and the statistics is averaged over 64 cores. The two particle Green s functions are sampled over a larger number of cores (4-5) to improve the statistical error bars. We sample the local two-particle Green s functions with CTQMC for all the correlated orbitals and compute the local polarization bubble to solve the inverse Bethe-Salpeter equation (BSE) for the local irreducible vertex. Finally, we compute the non-local polarization bubble G(k, ω)g(k Q, ω Ω) and combined with the local irreducible vertex (4) we obtain the full non-local spin and charge susceptibilities χ s,c (Q, Ω). The susceptibilities are computed on a Q-mesh. 1

11 D (ev -1 ) GW 8 4.3% 1.% LDA ω (ev) Figure 1: Density of States: Top pair of curves show density of states (DOS) from QSGW, for strains ɛ x =.3% (blue) and 1.% (green). A Van-Hove singularity appears slightly above the Fermi level when ɛ x =. When ɛ x the singularity splits into two peaks, with the smaller peak crossing E F at ɛ x =.6%. An intriguing possibility is to find conditions that cause the larger peak to cross E F. Bottom pair is the corresponding DOS in the LDA. The Van-Hove singularity and its splitting are also seen, but the peaks split more symmetrically, and evolve more slowly with ɛ x. On average, the LDA DOS is 5% smaller than the QSGW DOS, which is a consequence of the LDA s tendency to overestimate d bandwidths. Inset shows the QSGW DOS at ɛ x =.3% on finer energy scale. The grey line shows the DOS with spin-orbit coupling removed. 11

12 M Γ M ε=% ε=.3% ε=.6% ε=.9% M Γ M ε=.% ε=.3% ε=.6% ε=1.8% Figure : Fermi surfaces: Top row shows the QSGW Fermi surface in the basal plane, for a [1] tensile strain with ɛ x =(%,.3%,.6%,.9%). Spin orbit coupling is included (its omission makes a modest change to the Fermi surfaces). The four corners in the unstrained case correspond to the four M points q=(±1,, )π/a and q=(, ±1, )π/a; points marked M become distinct from the other two corners when ɛ x. Three bands cross E F. On the Γ-M line, the blue and green contours have predominately yz and xy symmetry; on Γ-X, blue, green, and red correspond mostly to xy, xz, and yz symmetry. Note change in topology along the Γ-M line at ɛ x=.6%. Bottom row shows corresponding results for DFT. In DFT the transition occurs near ɛ x =1.8% (bottom right panel), instead of.6%. 1

13 . x=1.%;, ev z x=.6%;, ev x=.%;, ev d xy M d yz M d xz M Figure 3: Non-local correlations: Quasi-particle renormalization factor, Z k (ω), computed in QSGW, as a heat map in the k ω plane. White lines detail Z at the Fermi energy (ω = ) along the Γ-M line. The three bands present at E F are all of Ru t g character, respectively d xy, d yz, d xz -like on this line (from left to right). Z is a measure of how strongly nonlocality in space and time renormalize and smear out the energy bands: at Z=1 electrons are perfectly coherent and act like independent particles; when Z= all coherence is lost. Its k-dependence is unusually strong. To put it in perspective, in iso-structural La CuO 4, Z varies by % (1) for the Cud x y band. Lower panels show Z k evolves in a non-trivial, orbital-dependent fashion with strain. Spin fluctuations missing in QSGW further reduce Z (Fig. 9). 13

14 q z = a/c x =.% q z = a/c x =.6% q z = a/c x = 1.% 8 Im( ), ev 1 7 mev q z = a/4c x =.% q z = a/4c x =.6% q z = a/4c x = 1.% mev 16 1 q z = x =.% q z = x =.6% q z = x = 1.% mev (,) (1/,1/) (1,1) (,) (1/,1/) (1,1) (,) (1/,1/) (1,1) Figure 4: Spin fluctuations: Imaginary part of the dynamic spin susceptibility χ s (q, ω) are shown in the Cartesian xy plane at different values q z, and for different strains ɛ x. The unstrained compound shows a spin fluctuation spectrum strongly peaked at (.3,.3, q z ) (units π/a). At ɛ x =, χ s is nearly independent of q z, but it begins to depend on q z for ɛ x >. With increasing strain fluctuations become more coherent and strongly peaked, reaching a zenith at ɛ x =ɛ x (.6%), where T c is maximum. For ɛ x >ɛ x, this peak becomes more diffuse; also a secondary incoherent peak emerges at (.15,.15, q z ), and the quasi anti-ferromagnetic vector (1/, 1/, a/c) acquires spectral weight around ω=4 mev. Note also the spectral weight near the FM vector (,,), and its evolution with ɛ x. 14

15 x, % x, % x, % x, % xy xy, ev 1 xy yz, ev 1 xy xz, ev 1 xz xz, ev 1 (,,) (1/,1/,) (1,1,) Figure 5: Charge fluctuations: Real part of the static charge susceptibility χ c (q, ω = ), shown along the Cartesian () to (11) direction, and for different strains ɛ x. The unstrained compound shows three-peaked charge fluctuation, with sharp peaks at IC vector (.,., ) (and by symmetry at (.8,.8, )) and a broad peak at (.5,.5, ). With strain the structure becomes sharply single-peaked at commensurate (.5,.5, ). The peak at the commensurate vector develops at the cost of the charge fluctuation weigts from the IC vectors. The systematic evolution from large wavelength triplet to shorter wavelength singlet flcutuations, under strain, is common to all inter- and intra-orbital charge flcutuations. The strong, often the most dominant, inter-orbital charge flcutuations can be observed in Ru-d xy xz and Ru-d xy,yz channels. 15

16 q y q y strain = 1.% strain =.6% q y q x q x q x q x strain =.% q x Figure 6: Pairing fields: The superconducting pairing gap symmetries for the leading five eigenvalues are shown for different strains ɛ x in the (1)-(1) plane in crystal coordinates. Nodal lines, gap minima and anti-nodal lines can be observed across all three spatial directions. For the unstrained case, the nodal lines, which appear as diagonal lines in the top row, correspond to the Cartesian q z -direction. The nodal structure evolves in a non-trivial fashion under strain, keeping the basic three dimensional gap structure intact across all strains. The first two eigenvalues for all strains are nearly degenerate. 16

17 Supplemental information Sr RuO 4 has a BCT lattice structure. We took the lattice parameters and internal positions from (41). The lattice constant in the basal plane is a= Å. The unaxial shear was performed at constant volume. 17

18 pz dyz dxz 68 dxy strain:.3%.6% py.9% px 8 7 dyz dxz dxy pz strain:.3%.6%.9% Figure 7: Energy levels: Relative orientations of different active orbitals in Sr RuO 4 from DFT and QSGW are shown. In strained Sr RuO 4, typical splitting of 4-7 mev between Ru-d xz and d yz orbitals and O-p x and O-p y are generic to both DFT and QSGW. This energy splitting is typically larger than the spin exchange scale in Sr RuO 4. However, orientations of active orbitals and the non-monotonic nature of their evolution across the critical point at.6% strain are very different from DFT, when non-local longer range charge correlations are treated with QSGW. 18

19 Γ M X M Γ X Figure 8: Band Structure and Spectral Functions: DFT, QSGW band structures and QSGW+DMFT (at 193 K) spectral functions are shown. 19

20 strain:. strain:.6 strain: 1. strain:. strain:.6 strain: 1. d xy T[K] d yz m*/m m*/m d xy strain:. strain:.6 strain: 1. T[K] d yz strain:. strain:.6 strain: T[K] d xz 3..5 T[K] strain:. strain:.6 strain: 1. m*/m 5 4 d xz strain:. strain:.6 strain: T[K] T[K] Figure 9: Scattering Rate and quasi-particle renormalization: The imaginary part of the QSGW+DMFT local self energy on Matsubara axes is fit to a fourth order polynomial at low energies, ImΣ(iω) = a + b(iω) + c(iω) + d(iω) 3 + e(iω) 4. The linear term is connected to the local mass enhancement m /m = b + 1, where m is the QSGW effective mass, and the intercept yields Γ through a = Γm /m. Γ (shown in ev) is the quasiparticle scattering rate at the lowest energy (4, 43). Note how strain modifies m /m and Γ for the xz and yz orbitals in a very different manner. This low-temperature mass anomaly is observed in Sr RuO 4 and discussed in previous studies (16, 18, 1).

21 q y q y strain =.% strain =.6% q y q x q x q x q x strain = 1.% -.1 val unit -.15 q x Figure 1: Pairing fields (Without SO): The superconducting pairing fields in five leading eigenvalues for different strains. While the details of the nodal and anti-nodal lines change from the with SO scenario, they still remain three dimensional. The eigenvalue spectrum is equally dense as the with SO case. 1

22 1 Ru 44 1/T 1, Hz strain O 8 1/T 1, Hz strain T, K Figure 11: Nuclear spin-relaxation rate: Nuclear spin relaxation rate 1 T 1 computed for Ru 44 and O 8 in the normal phase of Sr RuO 4. Both behave nearly linear in temperature, which is typical feature of a Fermi liquid. Our observed behavior of 1 T 1 in the normal phase agrees well with the experimental observations (44, 45, 46).

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