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Early proposals have suggested that superconductivity is dominant in the γ band. In particular, the candidate [[chiral p-wave]] order parameter in the momentum space exhibits k-dependence phase winding which is characteristic of time-reversal symmetry breaking. This peculiar single-band superconducting order is expected to give rise to appreciable spontaneous supercurrent at the edge of the sample. Such an effect is closely associated with the topology of the Hamiltonian describing Sr<sub>2</sub>RuO<sub>4</sub> in the superconducting state, which is characterized by a nonzero [[Chern number]]. However, scanning probes have so far failed to detect expected time-reversal symmetry breaking fields generated by the supercurrent, off by orders of magnitude.<ref name=Hicks2010>{{cite journal|last=Hicks|first=Clifford W.|title=Limits on superconductivity-related magnetization in Sr<sub>2</sub>RuO<sub>4</sub> and PrOs<sub>4</sub>Sb<sub>12</sub> from scanning SQUID microscopy|journal=Physical Review B|year=2010|volume=81|issue=21|pages=214501|doi=10.1103/PhysRevB.81.214501|arxiv = 1003.2189 |bibcode = 2010PhRvB..81u4501H |s2cid=26608198|display-authors=etal}}</ref> This has led some to speculate that superconductivity arises dominantly from the α and β bands instead.<ref name=Raghu2010>{{cite journal|last1=Raghu|first1=S.|last2=Marini|first2=Aharon|last3=Pankratov|first3=Steve|last4=Rubio|first4=Angel |title=Hidden Quasi-One-Dimensional Superconductivity in Sr<sub>2</sub>RuO<sub>4</sub>|url=http://prl.aps.org/abstract/PRL/v105/i13/e136401|journal= Physical Review Letters|volume=105|issue=13| page=136401|year=2010|arxiv = 1003.3927 |bibcode = 2010PhRvL.105b6401B |doi = 10.1103/PhysRevLett.105.026401|pmid=20867720|s2cid=26117260}}</ref> Such a two-band superconductor, although having k-dependence phase winding in its order parameters on the two relevant bands, is topologically trivial with the two bands featuring opposite Chern numbers. Therefore, it could possibly give a much reduced if not completely cancelled supercurrent at the edge. However, this naive reasoning was later found not to be entirely correct: the magnitude of edge current is not directly related to the topological property of the chiral state.<ref>{{cite journal | last1=Huang | first1=Wen | last2=Lederer | first2=Samuel | last3=Taylor | first3=Edward | last4=Kallin | first4=Catherine|author4-link= Catherine Kallin | title=Nontopological nature of the edge current in a chiralp-wave superconductor | journal=Physical Review B | volume=91 | issue=9 | date=2015-03-12 | issn=1098-0121 | doi=10.1103/physrevb.91.094507 | page=094507| arxiv=1412.4592 | bibcode=2015PhRvB..91i4507H | doi-access=free }}</ref> In particular, although the non-trivial topology is expected to give rise to protected chiral edge states, due to U(1) symmetry-breaking the edge current is not a protected quantity. In fact, it has been shown that the edge current vanishes identically for any higher angular momentum chiral pairing states which feature even larger Chern numbers, such as chiral d-, f-wave etc.<ref>{{cite journal | last1=Huang | first1=Wen | last2=Taylor | first2=Edward | last3=Kallin | first3=Catherine | title=Vanishing edge currents in non-p-wave topological chiral superconductors | journal=Physical Review B | volume=90 | issue=22 | date=2014-12-19 | issn=1098-0121 | doi=10.1103/physrevb.90.224519 | page=224519| arxiv=1410.0377 | bibcode=2014PhRvB..90v4519H | s2cid=118773764 }}</ref><ref>{{cite journal | last1=Tada | first1=Yasuhiro | last2=Nie | first2=Wenxing | last3=Oshikawa | first3=Masaki | title=Orbital Angular Momentum and Spectral Flow in Two-Dimensional Chiral Superfluids | journal=Physical Review Letters | volume=114 | issue=19 | date=2015-05-13 | issn=0031-9007 | doi=10.1103/physrevlett.114.195301 | page=195301| pmid=26024177 | arxiv=1409.7459 | bibcode=2015PhRvL.114s5301T | s2cid=3152887 }}</ref>
T<sub>c</sub> seems to increase under uniaxial compression<ref>{{cite journal|last1=Steppke|first1=Alexander|last2=Zhao|first2=Lishan|last3=Barber|first3=Mark E.|last4=Scaffidi|first4=Thomas|last5=Jerzembeck|first5=Fabian|last6=Rosner|first6=Helge|last7=Gibbs|first7=Alexandra S.|last8=Maeno|first8=Yoshiteru|last9=Simon|first9=Steven H.|last10=Mackenzie|first10=Andrew P.|last11=Hicks|first11=Clifford W.|date=2017-01-12|title=Strong peak in T<sub>c</sub> of Sr<sub>2</sub>RuO<sub>4</sub> under uniaxial pressure|url=http://pure-oai.bham.ac.uk/ws/files/102786770/Hicks_2017_Science.pdf|journal=Science|publisher=American Association for the Advancement of Science (AAAS)|volume=355|issue=6321|page=eaaf9398|doi=10.1126/science.aaf9398|issn=0036-8075|pmid=28082534|hdl-access=free|hdl=10023/10113|s2cid=8197509}}</ref> that pushes the [[van Hove singularity]] of the d<sub>xy</sub> orbital across the Fermi level.<ref>{{Cite journal|last1=Sunko|first1=Veronika|last2=Abarca Morales|first2=Edgar|last3=Marković|first3=Igor|last4=Barber|first4=Mark E.|last5=Milosavljević|first5=Dijana|last6=Mazzola|first6=Federico|last7=Sokolov|first7=Dmitry A.|last8=Kikugawa|first8=Naoki|last9=Cacho|first9=Cephise|last10=Dudin|first10=Pavel|last11=Rosner|first11=Helge|date=2019-08-19|title=Direct observation of a uniaxial stress-driven Lifshitz transition in Sr<sub>2</sub>RuO<sub>4</sub>|url=https://www.nature.com/articles/s41535-019-0185-9|journal=
In August 2021 evidence was reported for ''p''-wave singlet state as in cuprates and conventional superconductors, instead of the conjectured more unconventional ''p''-wave triplet state.<ref>{{Cite journal|last1=Chronister|first1=Aaron|last2=Pustogow|first2=Andrej|last3=Kikugawa|first3=Naoki|last4=Sokolov|first4=Dmitry A.|last5=Jerzembeck|first5=Fabian|last6=Hicks|first6=Clifford W.|last7=Mackenzie|first7=Andrew P.|last8=Bauer|first8=Eric D.|last9=Brown|first9=Stuart E.|date=2021-06-22|title=Evidence for even parity unconventional superconductivity in Sr<sub>2</sub>RuO<sub>4</sub>
==References==
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