Even odder after twenty-three years: the superconducting order parameter puzzle of Sr2RuO4

In this short review, we aim to provide a topical update on the status of efforts to understand the superconductivity of Sr2RuO4. We concentrate on the quest to identify a superconducting order parameter symmetry that is compatible with all the major pieces of experimental knowledge of the material, and highlight some major discrepancies that have become even clearer in recent years. As the pun in the title suggests, we have tried to start the discussion from scratch, making no assumptions even about fundamental issues such as the parity of the superconducting state. We conclude that no consensus is currently achievable in Sr2RuO4, and that the reasons for this go to the heart of how well some of the key probes of unconventional superconductivity are really understood. This is therefore a puzzle that merits continued in-depth study.


Introduction
The purpose of this short review is to give a status report on research into the superconducting properties, and most specifically the order parameter, of the widely-studied superconductor Sr2RuO4 1 . Our approach will be to remain open to all possibilities, and our conclusion will be that the issue is not settled after over twenty years of research. That being the case, it is perhaps worth beginning with a brief discussion of why this is an important problem, worthy of continued research.
Arguably the defining property of a so-called unconventional superconducting state is that the superconducting order parameter has a non-uniform phase in momentum space, such that it can be destroyed by sufficiently strong scattering from non-magnetic disorder 2 . The strength of scattering required depends on the strength of the superconductivity, and Sr2RuO4 has the most stringent purity criterion for observation of any known superconductor 3 . Its study therefore motivated the growth of extremely high quality single crystals 4 , in which it has been possible to determine the Fermi surface and normal state Fermi liquid quasiparticle properties with high accuracy and precision [5][6][7][8] . That Fermi surface is relatively simple. It consists of three sheets originating from three 4d orbitals of Ru with some contribution from the 2p orbitals of oxygen, and is highly twodimensional. In that sense it is slightly more complicated that the Fermi surfaces of the simplest unconventional superconductors (overdoped cuprates and some organic superconductors), but considerably simpler that those of many heavy fermion or pnictide superconductors. It has therefore been amenable to the construction of accurate but tractable tight-binding models, allowing the a host of modern many-body calculations to be compared with the properties of a real material [9][10][11][12][13][14][15][16][17][18] .
When one looks at the increasing sophistication of the techniques available for the study of unconventional superconductors, one has the feeling that the Sr2RuO4 problem really ought to be soluble, for several reasons. Firstly, the The differences between the theoretical predictions that have been made concerning Sr2RuO4 (which to some extent depend on the input assumptions made) are arguably less important than the common features that have emerged.
The most striking of these is illustrated in Fig. 1a for a calculation based on the model of Ref. 33 : Spin-fluctuation theories based on realistic parameterizations of the experimental Fermi surface and mass renormalisations of Sr2RuO4 find that the free energy difference between odd and even parity states is small.
Depending on the input parameters, either parity can be favoured, and among the richer odd parity states, there are also a number of near degeneracies. We stress this point because it immediately illustrates why determining the order parameter symmetry of Sr2RuO4 is not a trivial problem. Its physical origin # In the absence of spin-orbit coupling a pure spin triplet superconductor is odd parity, and a pure spin singlet superconductor is even parity. almost certainly lies in the structure of χ(q,ω). Although difficult to measure with precision, the similarity of the electronic structure of Sr2RuO4 to that of the itinerant ferromagnets SrRuO3 and Sr4Ru3O10 and the strongly enhanced metamagnet Sr3Ru2O7 indicates the likelihood of enhanced susceptibility near q = 0, a conclusion strengthened by the experiments showing that one of its Fermi surface sheets comes close to van Hove singularities at the M point of the twodimensional Brillouin zone [40][41][42] . Some broad weight is seen at low q in inelastic neutron scattering, but those experiments also famously established the existence of a prominent feature at approximately q = (2π/3a, 2π/3a) 43,44 . As might be expected of such an electronic structure 9 , the addition of significant levels of dopants such as Ti can stabilize static order at finite q 45 . Crudely speaking, a susceptibility with this kind of q structure can be exploited by many different flavours of spin-fluctuation mediated pairing, so it naturally places Sr2RuO4 close to the border between odd and even parity superconducting states.
The second notable feature, illustrated in Figs 1b and 1c, is the complexity of the predicted gap structures. Even the relatively simple Fermi surface of Sr2RuO4 introduces considerable variation in the average gap magnitude both between sheets and within a single sheet. Odd parity states may or may not have symmetry-imposed gap nodes, but the ones without nodes have deep gap minima, and the even parity states have a far richer nodal structure than one's naïve expectation based on experience of single-band superconductors.
Depending on one's point of view, Nature is either being unkind or kind hereunkind because of the near degeneracies among different order parameters and the complexity of the gap structures associated with those order parameters make the problem unexpectedly hard, or kind because it offers the prospect of rich superconducting phase diagrams, possibly including transitions between odd and even parity states.

Identification of key experiments
It is clear from the above discussion that unambiguous determination of the order parameter symmetry of Sr2RuO4 is likely to require accurate and precise experimental information, because there is not a sufficiently clear difference between the free energies of different candidate states for theory alone to provide a definitive answer. However, the calculations provide guidance on the classes of experiment that are likely to be the most important. Examination of Fig. 1 immediately suggests that thermodynamic data are likely to be complicated, showing signatures beyond those expected of a single gap 46 , and this is seen in experiment (Fig. 2). Even qualitative analysis of the temperature dependent heat capacity gives evidence for two or more gaps differing in magnitude by only of order a factor of two 47,48 . The second thing that Fig. 1 suggests is that measurements sensitive to the density of states in the vicinity of gap nodes will need to be performed under extremely stringent conditions if they are to yield definitive information. Ideally, they will need to go to extremely low temperatures (50 mK or below), be performed on the highest purity samples and have the capability of distinguishing accidental nodes or deep gap minima from those imposed by symmetry 49 . Considerable detail and very low temperature measurement will likely be required in order to distinguish one candidate order parameter from another.
The situation outlined above highlights the importance of measurements that are directly sensitive to symmetry. Admirable attempts have been made to conduct parity-sensitive tunneling studies of Sr2RuO4 50,51 ; while these have generally favoured odd parity superconducting states, the reproducibility from sample to sample is not as good as one would wish, so the results are better regarded as being suggestive than conclusive. There has also been an intriguing observation consistent with the existence of half flux quantum vortices in certain special conditions, again interpreted in terms of an odd parity state with a twocomponent order parameter, but not yet representing conclusive proof of such a state 52 . Another approach, still in its infancy but holding considerable promise, is the study of the proximity effect between Sr2RuO4 and metallic magnets 53 , for which the predicted behavior for odd and even parity states is substantially different 54 .

Experiments probing time reversal symmetry breaking
Considerable experimental effort has gone into an explicitly symmetry-related In spite of the above-mentioned caveats, the prevailing inference from the μSR and Kerr rotation experiments is that the observations result from the order parameter having two degenerate components in its 'orbital' degree of freedom § . If this is true, there are important consequences for the likely parity of the superconducting state, because not all candidate order parameter components ª In this context we note that there a large quantitative discrepancy between the size of the 0.5 G volume-averaged internal fields seen in the muon spin rotation measurements 55 and the much lower limit (≤ 1 mG) on internal fields established by scanning SQUID measurements 57-59 . § The possibility that the TRS-breaking might be in the 'spin' degree of freedom has not been widely investigated, though note the caveat above about the difficulties of even using this language in the presence of strongly k-dependent spin-orbit coupling effects. Tutorial-style descriptions of how to deduce the symmetry-breaking properties of different odd parity order parameters in the absence of spin-orbit coupling can be found in refs. 3,20 are degenerate in the absence of externally applied fields. The potential significance of this statement can be illustrated by considering the case of a material without spin-orbit coupling. In the tetragonal crystal field of Sr2RuO4 the only non-accidental way to have two degenerate d-wave order parameter components involving intra-band pairing is for them to be dxz and dyz 31 , but a TRS-breaking order parameter of the form dxz ± idyz would feature horizontal line nodes and Cooper pairs formed between electrons in different Ru-O planes § .
Although not impossible, and indeed also discussed theoretically in the context of odd parity order parameters 12,34 , interplane pairing would be a truly exotic state that seems intuitively unlikely in a material with such a strongly twodimensional Fermi surface . In contrast, p-wave components remain degenerate in a tetragonal crystal field, which is why a state of the form px ± ipy with in-plane pairing has been so extensively discussed in the literature.
One of the expectations of a simple px ± ipy state is the existence of edge currents which would produce measurable edge magnetic fields. Extensive experimental searches for these edge fields have yielded mostly null results [57][58][59]61 , but in the meantime more sophisticated calculations have suggested a variety of ways in which the edge currents could be far smaller that those predicted by the first naïve estimates [62][63][64][65] . More work will be needed to settle this issue completely, but for now it seems as if the lack of observed edge currents does not rule out the existence of a TRS breaking superconducting order parameter in Sr2RuO4.
Another consequence of a two-component order parameter might be the formation of domains in the superconducting state (though we note the § As stressed throughout this article, spin-orbit coupling is important in Sr2RuO4, so examination of the degeneracy-splitting of even parity order parameters based on in-plane Cooper pairing requires explicit numerical calculation using realistic multi-band models rather than simple estimates regarding a purely orbital part of a spin-orbit separable state. Such calculations confirm that the degeneracy splitting of the even parity states is usually substantial: for example, for the parameters used to produce Fig. 1 from the model of Ref. 33 , the predicted Tc of a dxy state is approximately one fifth of that of a " # $% # state. In the presence of interactions (included in the model 33 used to construct Fig. 1), these energetic differences become parameter-dependent, and accidental crossings can occur at which different even parity states involving in-plane become degenerate. One can also construct time-reversal-symmetry-breaking even parity states on the three-sheet Fermi surface of Sr2RuO4 involving inter-orbital pairing, but at the cost that in such states the intra-orbital pairing amplitude would have to be zero. The accurate statement, therefore, is that TRS-breaking condensates of even parity and in-plane Cooper pairs are not impossible, but would require fine-tuning to particular points in parameter space or the imposition of pairing conditions that both seem unlikely in a real material. comments on this in Ref. 66 ). A number of observations are qualitatively consistent with such a hypothesis 51,56,67,68 , therefore seemingly favouring the existence of an odd parity order parameter, but the estimates of the characteristic sizes of such domains vary widely.

Cooper pair formation and spin susceptibility in the superconducting state
One of the predictions for a simple even parity superconductor with weak spinorbit coupling is a strong drop in its spin susceptibility as the superconducting state is entered. This occurs because the non-magnetic singlet Cooper pairs are removed from the reservoir of conduction electrons whose energy can be lowered by field-induced spin polarization 69

Apparently contradictory results
If the TRS-breaking and spin susceptibility measurements were the only information available about Sr2RuO4, there would be little doubt that it has an odd parity order parameter. In reality, however, other work favours different conclusions. One prediction for px ± ipy (and for dxz ± idyz) is that lifting the tetragonal point-group symmetry of the system through in-plane magnetic field 76 or uniaxial pressure 77 should split the transition temperatures of the px and py components, yielding a double transition. Independent of microscopic detail, the splitting should be proportional to the strength of the applied symmetrybreaking field 77 . However experiments with both in-plane field 78,79 and uniaxial pressure 80,81 have not revealed such splitting.
Arguably an even more worrying discrepancy is revealed by studies of the superconducting upper critical field. In any superconductor, if the energy cost in maintaining equilibrium diamagnetism in the presence of the applied field becomes too high, the superconductivity is lost. It is well known that in standard spin singlet superconductors, contributions to this energy cost come from both creating the diamagnetic response and from a loss of spin energy in forming the Cooper pairs.
If the dominant energy cost comes from the energy required to expel the field by setting up appropriate screening currents, the critical field is often described as being 'orbitally limited'. As we shall see below, this terminology is confusing in the description of modern superconductors, so we will refer to the effect here as 'bulk diamagnetic orbital limiting'. In most superconductors this orbital limiting is dominant, and this is indeed the case for Sr2RuO4 with the magnetic field applied parallel to the crystallographic c-axis (H//c). If, however, the material is strongly type II, allowing efficient flux penetration via vortices, a second class of physics can limit the upper critical field if the Cooper pairs are spin singlets. As discussed above, spin singlet Cooper pairs are non-magnetic objects, so their formation results in a loss of magnetic polarization energy. If this loss is overbalanced by the superconducting condensation energy, the condensate forms and the spin susceptibility drops below Tc. As the applied field is raised, however, there comes a point at which the magnetic energy overpowers the condensation energy and the superconductivity is destroyed. This process is known as 'spin limiting' or 'Pauli limiting' 82 . For a fully-gapped spin singlet superconductor at T=0, the condition is: where χP is the Pauli susceptibility, H the applied field, N(0) the density of states at the Fermi level and Δ the superconducting energy gap. For this mechanism, the persistent currents giving the diamagnetic response disappear because the spin-related energetics result in the premature destruction of the Cooper pairs and hence of the condensate. In a multi-band material, appropriate averages need to be taken, but within factors of order one the prediction is that the limiting will occur when the applied field in tesla is the same as the transition temperature in kelvin. The Pauli limit is therefore a fundamental limit that can be observed in strongly type II superconductors with even parity order parameters. In simple interpretations, it should be entirely consistent with the information obtained from the Knight shift, because both phenomena are rooted in the competition between superconducting condensation energy and spin polarization energy, as illustrated in the sketch in Fig. 5.
For magnetic fields applied in the ab plane (H//ab), Sr2RuO4 is strongly type II. Its Fermi surface is so anisotropic that for this direction, the critical field based on bulk diamagnetic orbital limiting is expected to be at least a factor of 50 higher than that seen for the field applied parallel to c 3,6 . In fact, small angle scattering studies of the vortex lattice have established the intrinsic anisotropy parameter of the superconducting state, which determines the anisotropy of critical fields based on diamagnetic orbital limiting to be 60 83 . This would predict a critical field of approximately 4.5 T for H//ab. In contrast, the measured value is 1.5 T (Fig. 6a) 84 and the transition at low temperatures is first-order, as expected for Pauli limiting 85 rather than the second-order transition expected for diamagnetic orbital limiting [86][87][88]  This observation seems to be qualitatively at odds with the measurements of the NMR or neutron Knight shift, which give no evidence for a spin-related magnetic energy competing with the superconducting condensation energy (note the discrepancy between the combination of Figs. 4 and 6a and the sketch of Fig. 5).
The discrepancy has recently become even starker because of uniaxial pressure studies in which the Tc of Sr2RuO4 was raised to 3.5 K 42 . This was accompanied by an increase of the critical field for H//c by a factor of 20 from 0.075 T to 1.5 T.
Given the anisotropy factor of 60, an enormous critical field would then be predicted for H//ab, but instead only a modest rise to 4.5 T was observed (Fig.   6b).
The above observations make it seem certain that some critical field limiting mechanism is operating in Sr2RuO4. Quantitatively, it is in quite good agreement with the simple predictions of Pauli limiting associated with an even parity order parameter, but it is perhaps too early to jump firmly to that conclusion. For example, as pointed out in Ref. 89 , odd parity superconductors could themselves experience critical field limiting at some level in the presence of spin-orbit coupling. In these circumstances it is difficult to decouple a microscopic spin susceptibility from a microscopic orbital susceptibility, i.e. a magnetic susceptibility arising from the orbital character of the states at the Fermi surface.
The issue of how strong this effect could be in Sr2RuO4 will be a matter for precise calculation using models based on realistic parameterizations of the electronic structure including this spin-orbit coupling. Even if those calculations successfully accounted for the observed critical fields, however, other things would then need to be understood. In particular, it is urgent to obtain a ¨ In a simple weakly coupled superconductor with a uniform gap and no magnetic enhancement, the Pauli limit is that Hc2 in tesla should be a factor of 1.8 times Tc in kelvin 82 . Given the complexity of the gap and the magnetic susceptibility predicted for Sr2RuO4, direct numerical comparisons should be performed with extreme caution, so the fact that observed value is of order 1 instead of 1.8 should not be overinterpreted. theoretical understanding that reconciles critical field limiting of any microscopic origin with the fact that no associated reduction in susceptibility is observed in the NMR studies.

Summary and future work
The tone of the summary that one can give about the current situation is probably dependent on one's mood and one's natural levels of optimism or pessimism. The discrepancies that we have outlined above are not minor issues of detail but major qualitative disagreements between the results of experiments that are among the most prominent probes used in the study of superconductivity. At one level, these disagreements are a cause for depression about the state of the field of Sr2RuO4 physics (and certainly the source of conflict at conferences and meetings!). They also raise uncomfortable questions about how well we understand many other unconventional superconductors.
The issue is unlikely to be the quality of the experimental data. The data that we have focused on in this article are not the result of quick and speculative research on poorly-controlled samples, but stem from multiply-verified experiments conducted over many years on samples whose quality is among the highest available in any unconventional superconductor.
Instead, the contradictions observed in Sr2RuO4 suggest that interpretation of some of the key experiments commonly used in the field of unconventional superconductivity is not yet fully understood.
However, these problems can also be viewed as an opportunity. For all the reasons outlined in the introduction, Sr2RuO4 is a good superconductor on which to refine our understanding. It is therefore important that efforts to resolve the puzzles that it presents are continued and even stepped up. Although it is dangerous to try to predict the future in too much detail, several productive avenues of future research seem clear: i) There is much to be learned from the kind of studies pioneered in Ref. 53 of proximity effects between Sr2RuO4 and other magnetic and non-magnetic metals, with renewed efforts desirable on both the relevant experiment and theory.
ii) The existence of reliable and reproducible tunnel junctions into Sr2RuO4 would enable the parity-sensitive measurements that are so important to distinguish between putative classes of order parameter.
This research would likely receive a major boost if sufficiently pure thin films 90 could be grown reproducibly.
iii  95 give the hope that extension to low temperatures might become possible.
vi) It has been widely assumed that, with the best samples having mean free paths of microns and the coherence length being over one hundred times smaller, true clean-limit study of the phase diagram of Sr2RuO4 had been achieved. Very recently, however, it was reported in Ref. 96 that this might not be the case, with faint signs of superconductivity persisting to higher than expected applied fields in two extremely pure crystals. The authors of Ref. 96  inclusions also need to be carefully monitored, since there is considerable evidence that Tc is increased in their spatial vicinity 97,98 .
The recent observations on externally strained samples lead one to speculate that this Tc enhancement is due to internal strain, so obtaining truly pristine Sr2RuO4 will also likely require detailed knowledge and control of strain fields.
vii) The discussion in this review has largely focused on experiment, but there is also a clear need for continued work on theory, particularly on models concentrating on incorporating the k-dependent effects of spin-orbit coupling 37-39 in a realistic way. Is it possible that the contradictions outlined above only appear to be problems because the theories that are being used to frame the interpretation of the key experiments are still missing something? One obvious deficiency in most current theories based on 'realistic' electronic structure is that they are constructed in two dimensions, and hence ignore dispersion and spin-orbit coupling effects that vary with kz 37 . This is especially glaring in light of the strangeness of the properties in in-plane magnetic fields, so it is urgent that these theories be extended to the zdirection. Further theoretical work on collective modes 14,99 would also be desirable, as this leads to concrete predictions that can be investigated experimentally. In parallel with this, it would be interesting to continue to investigate the precise conditions required for the existence of topological superconductivity in Sr2RuO4 100 .
Overall, then, we prefer the optimistic point of view. In spite of the contradictions that exist in our current understanding of Sr2RuO4, the next decade of research on this fascinating material looks like being at least as exciting as the past two have been.

Figure from
Ref. 101 . Good agreement between experiment and theory (at least for temperatures above 100 mK) is not dependent on details since it only requires that the gaps on the two electronic subsystems be fairly similar in magnitude, and it does not matter whether the (alpha + beta) sheet gap or the gamma sheet gap is the larger one 48 . For completeness, the specific heat prediction for the odd parity gap structure of Fig. 1 b) is shown in the lower panel.  Extensive work has also been done using 17 O nuclei 70   is fully formed at low temperatures. If the applied field is low (left panel) this means that the spin susceptibility is quenched. However, if the applied field is sufficiently high, the spin polarisation energy gain wins out, and the superconductivity is destroyed. Although these sketches are for even parity superconductors in the absence of spin-orbit coupling, and the situation in Sr2RuO4 is much more complicated, the qualitative relationship between the two measurements would naively be expected to persist. It is thus a major discrepancy that critical field limiting is clearly seen in Sr2RuO4 but is not accompanied by a decrease of the Knight shift below Tc. Figure 6: Critical field limiting is seen in both Tc = 1.5 K Sr2RuO4 in ambient conditions (left panel) 86 and in strained Tc = 3.5 K material (right panel) 42 . In both cases there is evidence for a first order transition at low temperatures and high fields. This is a feature of simple theories of paramagnetic limiting in even parity superconductors 85 but might also be expected for magnetic limiting in spin-orbit coupled odd parity superconductors 89 .