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Register a new userStrain-induced time reversal breaking and half quantum vortices near a putative superconducting tetra-critical point in Sr$_2$RuO$_4$
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It has been shown [1] that many seemingly contradictory experimental findings concerning the superconducting state in Sr$_2$RuO$_4$ can be accounted for as resulting from the existence of an assumed tetra-critical point at near ambient pressure at which $d_{x^2-y^2}$ and $g_{xy(x^2-y^2)}$ superconducting states are degenerate. We perform both a Landau-Ginzburg and a microscopic mean-field analysis of the effect of spatially varying strain on such a state. In the presence of finite $xy$ shear strain, the superconducting state consists of two possible symmetry-related time-reversal symmetry (TRS) preserving states: $d pm g$. However, at domain walls between two such regions, TRS can be broken, resulting in a $d+ig$ state. More generally, we find that various natural patterns of spatially varying strain induce a rich variety of superconducting textures, including half-quantum fluxoids. These results may resolve some of the apparent inconsistencies between the theoretical proposal and various experimental observations, including the suggestive evidence of half-quantum vortices [2]. [1] Steven A Kivelson, Andrew C Yuan, BJ Ramshaw, and Ronny Thomale, A proposal for reconciling diverse experiments on the superconducting state in Sr$_2$RuO$_4$, npj Quantum Mater 5 (2020). [2] J Jang, DG Ferguson, V Vakaryuk, Raffi Budakian, SB Chung, PM Goldbart, and Y Maeno, Observation of half-height magnetization steps in Sr$_2$RuO$_4$, Science 331, 186-188 (2011).
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There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_text{TRSB}$, is generally found to match the critical temperature, $T_text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_text{TRSB} = T_text{c}$, but it has a hard-to-explain horizontal line node at $k_z=0$. Therefore, $s pm id$ and $d pm ig$ order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain $T_text{TRSB} approx T_text{c}$. To obtain evidence distinguishing these two possible scenarios (of symmetry-protected versus accidental degeneracy), we employ zero-field muon spin rotation/relaxation to study pure Sr$_2$RuO$_4$ under hydrostatic pressure, and Sr$_{1.98}$La$_{0.02}$RuO$_4$ at zero pressure. Both hydrostatic pressure and La substitution alter $T_text{c}$ without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected $T_text{TRSB}$ should track changes in $T_text{c}$, while if it is accidental, these transition temperatures should generally separate. We observe $T_text{TRSB}$ to track $T_text{c}$, supporting the hypothesis of $d_{xz} pm id_{yz}$ order.
Among unconventional superconductors, Sr$_2$RuO$_4$ has become a benchmark for experimentation and theoretical analysis because its normal-state electronic structure is known with exceptional precision, and because of experimental evidence that its superconductivity has, very unusually, a spontaneous angular momentum, i.e. a chiral state. This hypothesis of chirality is however difficult to reconcile with recent evidence on the spin part of the order parameter. Measurements under uniaxial stress offer an ideal way to test for chirality, because under uniaxial stress the superconducting and chiral transitions are predicted to split, allowing the empirical signatures of each to be identified separately. Here, we report zerofield muon spin relaxation (ZF-$mu$SR) measurements on crystals placed under uniaxial stresses of up to 1.05 GPa. We report a clear stress-induced splitting between the onset temperatures of superconductivity and time-reversal symmetry breaking, consistent with qualitative expectations for chiral superconductivity. We also report the appearance of unexpected bulk magnetic order under a uniaxial stress of ~ 1.0 GPa in clean Sr$_2$RuO$_4$.
Motivated by the success of experimental manipulation of the band structure through biaxial strain in Sr$_2$RuO$_4$ thin film grown on a mismatched substrate, we investigate theoretically the effects of biaxial strain on the electronic instabilities, such as superconductivity (SC) and spin density wave (SDW), by functional renormalization group. According to the experiment, the positive strain (from lattice expansion) causes charge transfer to the $gamma$-band and consequently Lifshitz reconstruction of the Fermi surface. Our theoretical calculations show that within a limited range of positive strain a p-wave superconducting order is realized. However, as the strain is increased further the system develops into the SDW state well before the Lifshitz transition is reached. We also consider the effect of negative strains (from lattice constriction). As the strain increases, there is a transition from p-wave SC state to nodal s-wave SC state. The theoretical results are discussed in comparison to experiment and can be checked by further experiments.
We analyze the spin anisotropy of the magnetic susceptibility of Sr$_2$RuO$4$ in presence of spin-orbit coupling and anisotropic strain using quasi-two-dimensional tight-binding parametrization fitted to the ARPES results. Similar to the previous observations we find the in-plane polarization of the low ${bf q}$ magnetic fluctuations and the out-of-plane polarization of the incommensurate magnetic fluctuation at the nesting wave vector ${bf Q}_1 = (2/3 pi ,2/3 pi)$ but also nearly isotropic fluctuations near ${bf Q}_2=(pi/6,pi/6)$. Furthermore, one finds that apart from the high-symmetry direction of the tetragonal Brillouin zone the magnetic anisotropy is maximal, i.e. $chi^{xx} eq chi^{yy} eq chi^{zz}$. This is the consequence of the orbital anisotropy of the $xz$ and $yz$ orbitals in the momentum space. We also study how the magnetic anisotropy evolves in the presence of the strain and find strong Ising-like ferromagnetic fluctuations near the Lifshitz transition for the $xy$-band.
We review electronic transport in superconducting junctions with Sr$_2$RuO$_4$. Transport measurements provide evidence for chiral domain walls and, therefore, chiral superconductivity in superconducting Sr$_2$RuO$_4$, but so far, the symmetry of the underlying superconducting state remains inconclusive. Further studies involving density of states measurements and spin-polarised transport in local/non--local Sr$_2$RuO$_4$ junctions with magnetic materials could lead to fundamental discoveries and a better understanding of the superconducting state.
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