We investigate pairing states realized at the (001) interface of a spin-triplet superconductor Sr$_2$RuO$_4$ on the basis of microscopic calculations. Because of a Rashba-type spin-orbit interaction induced at the interface, strong parity-mixing of Cooper pairs between a spin-singlet state and a spin-triplet state occurs in this system. There are also strong inter-band pair correlations between the spin-orbit split bands, in spite of the considerably large spin-orbit splitting. This is due to frustration between the spin-orbit interaction and pairing interactions. In this pairing state, time-reversal symmetry is restored, in contrast to the bulk Sr$_2$RuO$_4$ which is believed to be a chiral $p+ip$ superconductor with broken time-reversal symmetry. It is demonstrated that, because of these features, the pairing state at the interface is a promising candidate for the recently proposed time-reversal invariant topological superconductor.
Sr$_{2}$RuO$_{4}$ is one of the most promising candidates of a topological superconductor with broken time-reversal symmetry, because a number of experiments have revealed evidences for a spin-triplet chiral $p$-wave superconductivity. In order to clarify the time-reversal symmetry of Sr$_{2}$RuO$_{4}$, we introduce a novel test that examines the invariance of the Josephson critical current under the inversion of both the current and magnetic fields, in contrast to the detection of a spontaneous magnetic field employed in past experiments. Analyses of the transport properties of the planar and…
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 chi…
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…
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…
The internal structure of the superconducting state in Sr$_2$RuO$_4$ remains elusive at present, and exhibits evidence for time-reversal symmetry breaking. Recent muon spin relaxation measurements under uniaxial strain have revealed an increasing splitting between the superconducting critical temperature $T_c$ and the onset of time-reversal symmetry breaking $T_{\rm TRSB}$ with applied strain [Grinenko et al., arXiv:2001.08152]. In addition, static magnetic order is induced by the uniaxial strain beyond $\sim$1 GPa, indicating that unstrained Sr$_2$RuO$_4$ is close to a magnetic quantum critic…
The superconducting symmetry of Sr$_2$RuO$_4$ remains a puzzle. Time-reversal symmetry breaking $d_{x^2-y^2} + ig_{xy(x^2-y^2)}$ pairing has been proposed for reconciling multiple key experiments. However, its stability remains unclear. In this work, we theoretically study the superconducting instabilities in Sr$_2$RuO$_4$, including the effects of spin-orbit coupling (SOC), in the presence of both local and longer-range interactions within a random phase approximation. We show that the inclusion of second nearest neighbor repulsions, together with non-local SOC in the $B_{2g}$ channel or orbi…
Strontium ruthenate (Sr$_2$RuO$_4$) has long been thought to host a spin-triplet chiral $p$-wave superconducting state. However, the singletlike response observed in recent spin-susceptibility measurements casts serious doubts on this pairing state. Together with the evidence for broken time-reversal symmetry and a jump in the shear modulus $c_{66}$ at the superconducting transition temperature, the available experiments point towards an even-parity chiral superconductor with $k_z(k_x\pm ik_y)$-like $E_g$ symmetry, which has consistently been dismissed based on the quasi-two-dimensional electr…
We study the pairing symmetry of Sr$_2$RuO$_4$ through the group-theoretical approach. We emphasize the role of pairing interaction between the quasi-one-dimensional(Q1D) $d_{xz/yz}$ and quasi-two-dimensional(Q2D) $d_{xy}$ orbitals. It is found that two degenerate inter-orbital time-reversal-invariant(TRI) p-wave pairings, one is spin-singlet and the other spin-triplet with out-of-plane $\bm{d}$-vector, could be the most promising candidates. Several important physical quantities are presented, including the near-nodal gap structure, the unchanged out-of-plane Knight shift, and no split transi…
Origins of spontaneous magnetic fields in Sr$_2$RuO$_4$
PSI Center for Neutron and Muon Sciences CNM（瑞士保罗谢尔研究所中子与μ子科学中心）；STFC-ISIS Facility, Rutherford Appleton Laboratory（英国科学与技术设施委员会- ISIS设施，鲁瑟福阿普尔顿实验室）；National Institute for Materials Science（日本国立材料研究所）；Toyota Riken - Kyoto-Univ. Research Center (TRiKUC)（丰田理研-京都大学研究中心）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）
Muon spin rotation/relaxation ($μ$SR) and polar Kerr effect measurements provide evidence for a time-reversal symmetry breaking (TRSB) superconducting state in Sr$_2$RuO$_4$. However, the absence of a cusp in the superconducting transition temperature ($T_{\rm c}$) vs. stress and the absence of a resolvable specific heat anomaly at TRSB transition temperature ($T_{\rm TRSB}$) under uniaxial stress challenge a hypothesis of TRSB superconductivity. Recent $μ$SR studies under pressure and with disorder indicate that the splitting between $T_{\rm c}$ and $T_{\rm TRSB}$ occurs only when the structu…
Although correlated electronic-structure calculations explain very well the normal state of Sr$_2$RuO$_4$, its superconducting symmetry is still unknown. Here we construct the spin and charge fluctuation pairing interactions based on its correlated normal state. Correlations significantly reduce ferromagnetic in favor of antiferromagnetic fluctuations and increase inter-orbital pairing. From the normal-state Eliashberg equations, we find spin-singlet $d$-wave pairing close to magnetic instabilities. Away from these instabilities, where charge fluctuations increase, we find two time-reversal sy…
In this work, after reviewing the theory of the elastic properties of Strontium ruthenate, an extension suitable to explain the sound speed experiments of Lupien et. al. \protect\cite{lup2} and Clifford et. al. \protect\cite{clif1} is carried out. It is found that the discontinuity in the elastic constant C$_{66}$ gives unambiguous experimental evidence that the \sr superconducting order parameter $Ψ$ has two components and shows a broken time-reversal symmetry state. A detailed study of the elastic behavior is performed by means of a phenomenological theory employing the Ginzburg-Landau forma…
A comprehensive theory of superconductivity in Sr$_2$RuO$_4$ must explain experiments that suggest even parity superconducting order and others that suggest broken time reversal symmetry. Completeness further requires that the theory applies to Ca$_2$RuO$_4$, a Mott-Hubbard semiconductor that exhibits an unprecedented insulator-to-metal transition driven by very small electric field, and also by doping with very small concentration of electrons, leading to a metallic state proximate to ferromagnetism. A valence transition model, previously proposed for superconducting cuprates [Phys. Rev. B {\…
Due to the presence of a nontrivial three-dimensional spin-orbital entanglement, Sr$_2$RuO$_4$ may be a time-reversal invariant nematic $p$-wave superconductor with coexisting in-plane and out-of-plane pairings. Here we discuss various signatures of such a state if the out-of-plane pairing component is dominant. First, the enhancement of the superconducting $T_c$ under in-plane uniaxial strains is nearly a quadratic function of the strain, because the out-of-plane pairing lacks a linear-order coupling to the strain. Second, when the strain applies along a certain in-plane direction, the nemati…
The symmetry of the superconducting condensate in Sr$_2$RuO$_4$ remains controversial after nuclear magnetic resonance (NMR) experiments recently overturned the dominant chiral $p$-wave paradigm. Several theoretical proposals have been put forward to account for existing experiments, including a $d+ig$-wave admixture, conjectured to be stabilized by longer-range Coulomb interactions. We perform a material-specific microscopic theoretical study of pairing by spin- and charge-fluctuations in Sr$_2$RuO$_4$, including the effects of spin-orbit coupling, and both local and longer-range Coulomb repu…
The superconducting pairing in Sr$_2$RuO$_4$ is widely considered to be chiral $p$-wave with $\vec{d}_{\boldsymbol k} \sim (k_x + ik_y)\hat{z}$, which belongs to the $E_u$ representation of the crystalline $D_{4h}$ group. However, this superconducting order appears hard to reconcile with a number of key experiments. In this paper, based on symmetry analysis we discuss the possibility of odd-parity pairing with inherent three-dimensional (3D) character enforced by the inter-orbital interlayer coupling and the sizable spin-orbit coupling (SOC) in the material. We focus on a yet unexplored $E_u$ …
The superconducting state in the quasi-two-dimensional and strongly correlated Sr$_2$RuO$_4$ is uniquely held up as a solid state analog to superfluid $^3$He-$A$, with an odd-parity order parameter that also breaks time reversal symmetry, and for which the vector order parameter has the same direction in spin space for all electron momenta. The recent discovery that uniaxial pressure causes a steep rise and maximum in transition temperature ($T_c$) in strained samples motivated the study of $^{17}$O nuclear magnetic resonance (NMR) that we describe in this article. A reduction of Knight shifts…
Standard superconductors display a ubiquitous discontinuous jump in the electronic specific heat at the critical superconducting transition temperature. In a growing class of unconventional superconductors, however, a second order parameter component may get stabilized and produce a second heat capacity jump at a lower temperature, typically associated with the spontaneous breaking of time-reversal symmetry. The splitting of the two specific heat discontinuities can be controlled by external perturbations such as chemical substitution, hydrostatic pressure, or uniaxial strain. We develop a the…
Directional Andreev-Reflection Signatures of Inter-Orbital Pairing in Sr$_2$RuO$_4$
CNR-SPIN, c/o Università di Salerno（意大利萨莱诺大学CNR-SPIN研究所）；Faculty of Environmental Life, Natural Science and Technology, Okayama University（冈山大学环境生命理工学部）；Department of Applied Physics, Nagoya University（名古屋大学应用物理系）；Max-Planck Institute for Solid State Research（斯图加特马克斯·普朗克固体研究所）；University of St Andrews（圣安德鲁斯大学）；Department of Applied Science and Technology, Politecnico di Torino（都灵理工大学应用科学与技术系）
We present a systematic investigation of muon-stopping states in superconductors that reportedly exhibit spontaneous magnetic fields below their transition temperatures due to time-reversal symmetry breaking. These materials include elemental rhenium, several intermetallic systems and Sr$_2$RuO$_4$. We demonstrate that the presence of the muon leads to only a limited and relatively localized perturbation to the local crystal structure, while any small changes to the electronic structure occur several electron volts below the Fermi energy leading to only minimal changes in the charge density on…
The collective mode spectrum of a symmetry-breaking state, such as a superconductor, provides crucial insight into the nature of the order parameter. In this context, we present a microscopic weak-coupling theory for the collective modes of a generic multi-component time-reversal symmetry breaking superconductor, and show that fluctuations in the relative amplitude and phase of the two order parameter components are well-defined underdamped collective modes, even in the presence of nodal quasiparticles. We then demonstrate that these "generalized clapping modes" can be detected using a number …
The paper reviews recent work on time reversal symmetry (TRS) breaking superconductors. The family of TRS breaking superconductors is growing relatively fast, with many of its newly discovered members being non-centrosymmetric. However, many of the superconductors which possess center of inversion also break TRS. The TRS is often identified by means of the muon spin relaxation effect ($μ$SR) or/and the Kerr effect. Both methods effectively measure the spontaneous appearance of the bulk magnetic field below superconducting transition temperature. One of the systems most carefully studied so far…
Recent theoretical work predicted emergence of chiral topological superconducting phase with spontaneously broken time reversal symmetry in a twisted bilayer composed of two high-$T_c$ cuprate monolayers, such as Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$. Here we identify large intrinsic Hall response that can be probed through the polar Kerr effect measurement as a convenient signature of the $\mathcal{T}$-broken phase. Our modelling predicts the Kerr angle $θ_K$ to be in the range of 10-100 $μ$rad, which is a factor of $10^3-10^4$ times larger than what is expected for the leading chiral supercondutor c…
In general, magnetism and superconductivity are antagonistic to each other. However, there are several families of superconductors, in which superconductivity may coexist with magnetism, and only a few examples are known, when superconductivity itself induces spontaneous magnetism. The most known compounds are Sr$_2$RuO$_4$ and some noncentrosymmetric superconductors. Here, we report the finding of a narrow dome of a novel $s+is'$ superconducting (SC) phase with broken time-reversal symmetry (BTRS) inside the broad $s$-wave SC region of the centrosymmetric multiband superconductor Ba$_{\rm 1-x…
We analyze the presence of non-reciprocal critical currents, the so-called superconducting diode effect, in chiral superconductors within a generalized Ginzburg-Landau framework. After deriving its key symmetry conditions we illustrate the basic mechanism for two examples, the critical current in a thin film and a Josephson junction. The appearance of spontaneous edge currents and the energy bias for the formation of Josephson vortices play an essential part in establishing a splitting of the critical currents running in opposite directions. Eventually, this allows us to interpret a supercondu…
Chains of magnetic atoms, placed on the surface of s-wave superconductors, have been established as a laboratory for the study of Majorana bound states. In such systems, the breaking of time reversal due to magnetic moments gives rise to the formation of in-gap states, which hybridize to form one-dimensional topological superconductors. However, in unconventional superconductors even non-magnetic impurities induce in-gap states since scattering of Cooper pairs changes their momentum but not their phase. Here, we propose a path for creating topological superconductivity, which is based on an un…
The off-diagonal components of thermal conductance tensor, thermal Hall conductivities (THCs), have extensively been studied in recent condensed matter experiments to investigate fractionalized quantum spin liquids, and quantum Hall systems. Under zero magnetic field, THCs spontaneously become non-zero for time-reversal symmetry (TRS) broken systems, and can have contributions from topologically protected edge states. Here we focus on an additional bulk effect, the impurity mechanism in TRS broken superconductors. Inspired by $Sr_2 Ru O_4$, the low temperature THC was calculated [Sup. Sci. and…
Recent nuclear magnetic resonance studies [A. Pustogow {\it et al.}, arXiv:1904.00047] have challenged the prevalent chiral triplet pairing scenario proposed for Sr$_2$RuO$_4$. To provide guidance from microscopic theory as to which other pair states might be compatible with the new data, we perform a detailed theoretical study of spin-fluctuation mediated pairing for this compound. We map out the phase diagram as a function of spin-orbit coupling, interaction parameters, and band-structure properties over physically reasonable ranges, comparing when possible with photoemission and inelastic n…
We examine the stability of mixed-symmetry superconducting states with broken time-reversal symmetry in spatial-symmetry-broken systems, including chiral states, on the basis of the free-energy functional derived in the weak-coupling theory. We consider a generic a_1 + i a_2 wave state, with a_1 and a_2 being different symmetry indices such as (a_1,a_2) = (d,s), (p_x,p_y), and (d,d').The time-reversal symmetry of the mixed-symmetry state with the a_1- and a_2-wave components is broken when the phases of these components differ, and such a state is called the time-reversal-symmetry breaking (TR…
We consider weak-coupling superconductivity in the inversion- and rotation-symmetric system of two pseudospin $s=1$ low-energy fermions of opposite chirality. General contact interactions can lead to Cooper instabilities towards d-wave or towards a novel $2\times 3$ p-wave matrix order parameter. We compute the Ginzburg-Landau free energy for the latter. Remarkably, in this case the Ginzburg-Landau free energy can be minimized exactly, with the resulting ordered state being analogous to the ``axi-planar" p-wave, which exhibits extra degeneracy, and generally breaks time reversal symmetry. Wher…
It has been shown that singlet Cooper pairs can be converted into triplet ones and diffuse into a ferromagnet over a long distance in a phenomenon known as the long-range proximity effect (LRPE). This happens in materials with inhomogeneous magnetism or spin-orbit coupling (SOC). Most of the previous studies focus on the cases with small SOC and exchange field. However, the physics was not clear when SOC and exchange field strength are both much greater than the disorder strength. In this work, we consider a two dimensional system with a large Rashba-type SOC and exchange field in the case whe…
