The long-standing quest to determine the superconducting order of Sr$_2$RuO$_4$ (SRO) has received renewed attention after recent nuclear magnetic resonance (NMR) Knight shift experiments have cast doubt on the possibility of spin-triplet pairing in the superconducting state. As a putative solution, encompassing a body of experiments conducted over the years, a $d+ig$-wave order parameter caused by an accidental near-degeneracy has been suggested [S. A. Kivelson et al., npj Quantum Materials $\bf{5}$, 43 (2020)]. Here we develop a general Ginzburg--Landau theory for multiband superconductors. We apply the theory to SRO and predict the relative size of the order parameter components. The heat capacity jump expected at the onset of the second order parameter component is found to be above the current threshold deduced by the experimental absence of a second jump. Our results tightly restrict theories of $d+ig$ order, and other candidates caused by a near-degeneracy, in SRO. We discuss possible solutions to the problem.
We study the spin susceptibility of the orbital-singlet pairings, including the spin-triplet/orbital-singlet/$s$-wave $E_g$ representation proposed by Suh \textit{et al}., [H.\ G.\ Suh \textit{et al}., Phys.\ Rev.\ Research \textbf{2}, 032023 (2020)], for a three-orbital model of superconducting Sr$_2$RuO$_4$ in three dimensions. For the pseudospin-singlet states represented in the band basis, the spin susceptibility decreases when reducing the temperature, irrespective of the direction of the applied magnetic fields, even if they are spin-triplet/orbital-singlet pairings in the spin-orbital s…
Shortly after the discovery of superconductivity in Sr$_2$RuO$_4$, spin-triplet pairing was proposed and further corroborated by a constant Knight shift ($K$) across the transition temperature (T$_c$). However, a recent experiment observed a drop in $K$ at T$_c$ which becomes larger under uniaxial strain, ruling out several spin-triplet scenarios. Here we show that inter-orbital spin-triplet pairing features a d-vector that rotates when uniaxial strain is applied, leading to a larger drop in the spin polarization perpendicular to the strain direction, distinct from spin-singlet pairing. We pro…
We have measured the diffracted neutron scattering intensities from the square magnetic flux lattice in the perovskite superconductor Sr$_2$RuO$_4$, which is thought to exhibit p-wave pairing with a two-component order parameter. The relative intensities of different Bragg reflections over a wide range of field and temperature have been shown to be inconsistent with a single component Ginzburg-Landau theory but qualitatively agree with a two component p-wave Ginzburg-Landau theory.
We derive the Ginzburg-Landau theory of unconventional singlet superconductors in the presence of a Zeeman field and impurities, in order to examine the resulting Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. We show that the behavior of the FFLO phases in unconventional superconductors in the presence of impurities is qualitatively different from that found for s-wave superconductors.
We study the dynamical spin susceptibility, $χ({\bf q}, ω)$, in the normal and superconducting state of Sr$_2$RuO$_4$. In the normal state, we find a peak in the vicinity of ${\bf Q}_i\simeq (0.72π,0.72π)$ in agreement with recent inelastic neutron scattering (INS) experiments. We predict that for spin triplet pairing in the superconducting state a {\it resonance peak} appears in the out-of-plane component of $χ$, but is absent in the in-plane component. In contrast, no resonance peak is expected for spin singlet pairing.
The unconventional superconductivity in Sr$_2$RuO$_4$ continues to defy a unified interpretation. In this paper, we focus on some novel aspects of its superconducting pairing by exploiting the orbital degree of freedom in this material. The multi-orbital nature, combined with the symmetry of the orbitals involved, leads to a plethora of exotic Cooper pairings not accessible in single-orbital systems. Essential physics is illustrated first using a two-orbital model with $d_{xz}$- and $d_{yz}$-orbitals. We classify the gap functions according to the underlying lattice symmetries, analyze the eff…
Based on the Ginzburg-Landau functional of E_{u} symmetry presented by Agterberg, vortex states of Sr_{2}RuO_{4} are studied in detail over $H_{c1}\lesssim H \leq H_{c2}$ by using the Landau-level expansion method. For the field in the basal plane, it is found that (i) the second superconducting transition should be present irrespective of the field direction; (ii) below this transition, a characteristic double-peak structure may develop in the magnetic-field distribution; (iii) a third transition may occur between two different vortex states. It is also found that, when the field is along the…
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…
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…
The possibility of non-s-wave superconductivity induced by phonons is investigated using a simple model that is inspired by Sr$_2$RuO$_4$. The model assumes a two-dimensional electronic structure, a two-dimensional spin-fluctuation spectrum, and three-dimensional electron-phonon coupling. Taken separately, each interaction favors formation of spin-singlet pairs (of s symmetry for the phonon interaction and d$_{x^2-y^2}$ symmetry for the spin interaction), but in combination, a variety of more unusual singlet and triplet states are found, depending on the interaction parameters. This may have i…
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 {\…
We study theoretically a model for twin boundaries in superconductors with Rashba spin-orbit coupling, which can be relevant to both three-dimensional noncentrosymmetric tetragonal crystals and two-dimensional gated superconductors such as the LaAlO$_3$/SrTiO$_3$ interface. In both cases, the broken inversion symmetry allows for a coexistence of singlet and triplet pairing. Within the framework of a Ginzburg-Landau theory, we identify two $\mathbb{Z}_2$ symmetries that are broken via two consecutive second order phase transitions as the temperature is lowered. We show that a time-reversal symm…
Ginzburg-Landau (GL) equations for the coexistent states of superconductivity and magnetism are derived microscopically from the extended Hubbard model with on-site repulsive and nearest-neighbor attractive interactions. In the derived GL free energy a cubic term that couples the spin-singlet and spin-triplet components of superconducting order parameters (SCOP) with magnetization exists. This term gives rise to a spin-triplet SCOP near the interface between a spin-singlet superconductor and a ferromagnet, consistent with previous theoretical studies based on the Bogoliubov de Gennes method an…
Sr$_2$RuO$_4$ has been under intensive scrutiny over the past years after new NMR measurements unveiled that the superconducting state might be a spin singlet. One of the best order parameter candidates in light of these new experiments is a chiral d-wave state with $d_{xz}+id_{yz}$ symmetry. This order parameter has been overlooked given the strong two-dimensional character of the normal state electronic structure. Recently, a phenomenological proposal based on local interactions with a three-dimensional electronic structure showed that a chiral d-wave state can be stable in Sr$_2$RuO$_4$ onc…
The microscopic mechanism of circular currents induced in the vicinity of a non-magnetic impurity is analyzed for the (time-reversal symmetry breaking) chiral superconducting state ${\bf d}({\bf k}) = \hat{\bf z} (k_x \pm i k_y)$ which is very likely realized in Sr_2RuO_4. From the analytic, not self-consistent solution of the Bogolyubov-de Gennes equations we find two types of quasiparticle states, a bound state and the continuum state. Through impurity scattering the condensate transfers angular momentum to the quasiparticle states which generate a circular current. At low temperature the co…
We report a novel phenomenon intimately related to the spin-triplet superconductivity. It is well known that the spin susceptibility decreases below the superconducting transition temperature in almost all superconductors because of spin-singlet pair formation, while it may remain unchanged in a handful of spin-triplet exceptions. Here we report the observation in Sr$_2$RuO$_4$ with nuclear magnetic resonance (NMR) that the spin susceptibility originating from the Ru-4$d$ electron slightly $increases$ by $\sim 2 $\% of total and becomes inhomogeneous in the superconducting state. These are rea…
It has been shown [1] that many seemingly contradictory experimental findings concerning the superconducting state in Sr$_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…
Muon Knight shift as a precise probe of the superconducting symmetry of Sr$_2$RuO$_4$
Toyota Riken–Kyoto University Research Center (TRiKUC), Kyoto University（丰田理研-京都大学研究中心）；Institute for Chemical Research, Kyoto University（京都大学化学研究所）；PSI Center for Neutron and Muon Sciences CNM（保罗谢尔研究所中子与缪子科学中心）；Institute for Solid State Physics, the University of Tokyo（东京大学固体物理研究所）；Department of Physics, University of British Columbia（不列颠哥伦比亚大学物理系）；Department of Electronic Science and Engineering, Graduate School of Engineering, Kyoto University（京都大学大学院工学研究科电子科学与工程系）
The ruthenate family of layered perovskites has been a topic of intense interest with much work dedicated to understanding the superconducting state of the single layer, Sr$_2$RuO$_4$. Another longstanding puzzle is the lack of superconductivity in its sister compound, Sr$_3$Ru$_2$O$_7$, which constrains the possible superconducting mechanisms of Sr$_2$RuO$_4$. Here we address a microscopic mechanism that unifies superconducting and spin-density wave order in this family of materials. Beginning from a model of Sr$_2$RuO$_4$ featuring intraband pseudospin-singlet superconductivity originating f…
We investigate the order parameter of Sr$_2$RuO$_4$ in an exchange-only magnetic field. A Ginzburg-Landau symmetry analysis implies three possibilities: a pure $^3$He A phase, a $^3$He A$_1$ or a $^3$He A$_2$ phase. We explore the exchange field dependence of the order parameter and energy gap in a one-band model of Sr$_2$RuO$_4$. The numerical solutions show no A$_1$ phase and that the A$_2$ phase is lower in free energy than the A phase. We explore heat capacity as a function of temperature and field strength and find quantitatively different behaviors for the A and A$_2$ phases.
I investigate the vortex lattice structure of the Ginzburg Landau free energy for a two component order parameter in the weak-coupling clean-limit when the field is along the high symmetry axis in a tetragonal crystal. It is shown that the vortex lattice phase diagram as a function of the Ginzburg Landau free energy parameters includes phases with a hexagonal, centered rectangular, rectangular, and square unit cells. It is also shown that the square vortex lattice has the largest region of stability. The field distribution of the square vortex lattice near $H_{c2}$ is determined and the applic…
The magnetization and magnetic torque of a high-quality single crystal of Sr$_2$RuO$_4$ have been measured down to 0.1 K under a precise control of the magnetic-field orientation. When the magnetic field is applied exactly parallel to the $ab$ plane, a sharp magnetization jump $4πδM$ of $(0.74 \pm 0.15)$ G at the upper critical field $H_{{\rm c2},{ab}} \sim 15$ kOe with a field hysteresis of 100 Oe is observed at low temperatures, evidencing a first-order superconducting-normal transition. A strong magnetic torque appearing when $H$ is slightly tilted away from the $ab$ plane confirms an intri…
The data of temperature dependent superfluid density $n_s(T)$ in Li$_2$Pd$_3$B and Li$_2$Pt$_3$B [Yuan {\it et al.}, \phrl97, 017006 (2006)] show that a sudden change of the slope of $n_s (T)$ occur at slightly lower than the critical temperature. Motivated by this observation, we microscopically derive the Ginzburg-Landau (GL) equations for noncentrosymmetric superconductors with Rashba type spin orbit interaction. Cooper pairing is assumed to occur between electrons only in the same spin split band and pair scattering is allowed to occur between two spin split bands. The GL theory of such a …
We consider Fermions in two dimensions with an attractive interaction in the singlet d-wave channel of arbitrary strength. By means of a Hubbard-Stratonovich transformation a statistical Ginzburg-Landau theory is derived, which describes the smooth crossover from a weak-coupling BCS superconductor to a condensate of composite Bosons. Adjusting the interaction strength to the observed slope of H_c2 at T_c in the optimally doped high-T_c compounds YBCO and BSCCO, we determine the associated values of the Ginzburg-Landau correlation length xi and the London penetration depth lambda. The resulting…
Using Ginzburg-Landau formalism, we theoretically study the isothermal shear modulus anomaly of an unconventional superconductor with a multicomponent order parameter, when the superconducting transition is split by a symmetry-breaking field. Experimental signatures are proposed for both chiral and nematic superconductors. Particularly striking is the vanishing of $C_{66}$ across the lower transition to a nematic superconducting state. Our findings can guide future experiments and shed new lights on materials such as Sr$_2$RuO$_4$ and $M_x$Bi$_2$Se$_3$.
The possible superconducting states of strontium ruthenate (Sr$_2$RuO$_4$) are organized into irreducible representations of the point group $D_{4h}$, with a special emphasis on nodes occurring within the superconducting gap. Our analysis covers the cases with and without spin-orbit coupling and takes into account the possibility of inter-orbital pairing within a three-band, tight-binding description of Sr$_2$RuO$_4$. No dynamical treatment if performed: we are confining ourselves to a group-theoretical analysis. The case of uniaxial deformations, under which the point group symmetry is reduce…
Induction of p-wave and d-wave order parameters in s-wave superconductors with light pulses
Nordita, Stockholm University, and KTH Royal Institute of Technology（诺尔迪塔，斯德哥尔摩大学，以及皇家理工学院）；Department of Physics, University of Connecticut（康涅狄格大学物理系）
The close interplay between superconductivity and antiferromagnetism in several quantum materials can lead to the appearance of an unusual thermodynamic state in which both orders coexist microscopically, despite their competing nature. A hallmark of this coexistence state is the emergence of a spin-triplet superconducting gap component, called $π$-triplet, which is spatially modulated by the antiferromagnetic wave-vector, reminiscent of a pair-density wave. In this paper, we investigate the impact of these $π$-triplet degrees of freedom on the phase diagram of a system with competing antiferr…
High temperature superconductivity in the cuprates remains one of the most widely investigated, constantly surprising, and poorly understood phenomena in physics. Here, we describe briefly a new phenomenological theory inspired by the celebrated description of superconductivity due to Ginzburg and Landau and believed to describe its essence. This posits a free energy functional for the superconductor in terms of a complex order parameter characterizing it. We propose, for superconducting cuprates, a similar functional of the complex, in plane, nearest neighbor spin singlet bond (or Cooper) pai…
Soon after the discovery of superconductivity in Sr2RuO4 (SRO) in 1994, it was conjectured that its order parameter (OP) has a form similar to that realized in the superfluid He-3, namely, odd parity and spin triplet. The chiral p-wave pairing realized in He-3 A-phase was favored by several early experiments, in particular, the muon spin rotation and the Knight shift measurements published in 1998. However, the original Knight shift result was called into question in early 2019, raising the question as to whether the chiral p-wave, or even the spin-triplet pairing itself, is indeed realized in…
We propose a phenomenological Ginzburg-Landau-like theory of cuprate superconductivity. The free energy is expressed as a functional F of the spin-singlet pair amplitude psi_ij=psi_m=Delta_m exp(i phi_m); i and j are nearest-neighbor sites of the Cu lattice in which the superconductivity is believed to primarily reside and m labels the site at the center of the bond between i and j. The system is modeled as a weakly coupled stack of such planes. We hypothesize a simple form, F[Delta,phi]=sum_m (A Delta_m^2+ B Delta_m^4/2)+C sum_<mn> Delta_m Delta_n cos(phi_m-phi_n), for the functional. The coe…
The Josephson current between an s-wave and a spin-triplet superconductor Sr$_2$RuO$_4$ (SRO) is studied theoretically. In spin-singlet / spin-triplet superconductor junctions, there is no Josephson current proportional to $\sin ϕ$ in the absence of the spin-flip scattering near junction interfaces, where $ϕ$ is a phase-difference across junctions. Thus a dominant term of the Josephson current is proportional to $\sin 2ϕ$ . The spin-orbit scattering at the interfaces gives rise to the Josephson current proportional to $\cosϕ$, which is a direct consequence of the chiral paring symmetry in SRO.
Based on a theoretical argument and Monte Carlo simulations of a Ginzburg-Landau model derived microscopically, it is argued that, in type-II superconductors where {\it both} the paramagnetic {\it and} orbital depairings are important, a strong first-order transition (FOT) at $H_{c2}$ expected in the mean field (MF) approximation never occurs in real systems and changes due to the fluctuation into a crossover. The present result explains why a {\it nearly} discontinuous crossover at $H_{c2}$ with {\it no} intrinsic hysteresis is observed only in a clean superconducting material with a singlet …
For a tetragonal material, order parameters of $p_x$ and $p_y$ symmetry are related by rotation and hence have the same $T_{\rm c}$ at a mean-field level. This degeneracy can be lifted by a symmetry-breaking field, like (uniaxial) in-plane strain, such that at $T_{\rm c}$, the order parameter is only of $p_x$ or $p_y$ symmetry. Only at a lower temperature also the respective other order parameter condenses to form a chiral $p$-wave state. At the mean-field level, the derivative of $T_{\rm c}$ with strain is discontinuous at zero strain. We analyze consequences of (thermal) fluctuations on the …
A Ginzburg-Landau theory is developed for unconventional superconductors with the three relevant singlet pairing channels. Various consequences of the sub-dominant channels (i.e., s- and d_{xy}-channels) are examined in detail. (1) In the case of a d_{x^2-y^2}+is-wave superconductor, The structure of a single vortex above and below T_{DS} is four-fold and two-fold symmetric, respectively. (2) In the case of a d_{x^2-y^2}+id_{xy}-wave superconductor, there is also a second order zero-field phase transition from the pure d_{x^2-y^2}-phase to the Time-reversal-symmetry-breaking d_{x^2-y^2}+id_{xy…
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 investigate the magnetic field dependence of an ideal superconducting vortex lattice in the parity-mixed pair-density wave phase of multilayer superconductors within a circular cell Ginzburg-Landau approach. In multilayer systems, due to local inversion symmetry breaking, a Rashba spin-orbit coupling is induced at the outer layers. This combined with a perpendicular paramagnetic (Pauli) limiting magnetic field stabilizes a staggered layer dependent pair-density wave phase in the superconducting singlet channel. The high-field pair-density wave phase is separated from the low-field BCS phase…
In superconductors with singlet pairing, the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is expected to be stabilized by a large Zeeman splitting. We develop an efficient method to evaluate the Landau-Ginzburg free energies of FFLO-state vortex lattices and use it to simplify the considerations that determine the optimal vortex configuration at different points in the phasediagram. We demonstrate that the order parameter spatial profile is completely determined, up to a uniform translation, by its Landau level index n and the vortex Lattice structure and derive an explicit expr…
We propose a superconducting instability where microscopic supercurrent loops form spontaneously within a unit cell at the superconducting transition temperature with only uniform, onsite and intra-orbital singlet pairing. As a result of the circulating currents time-reversal symmetry is spontaneously broken in the superconducting state. Using Ginzburg-Landau theory, we describe in detail how these currents emerge in a toy model. We discuss the crystallographic symmetry requirements to realize such a state and show that they are met by the Re6X (X=Zr, Hf, Ti) family of time-reversal symmetry b…
