A time reversal invariant (TRI) topological superconductor has a full pairing gap in the bulk and topologically protected gapless states on the surface or at the edge. In this paper, we show that in the weak pairing limit, the topological quantum number of a TRI superconductor can be completely determined by the Fermi surface properties, and is independent of the electronic structure away from the Fermi surface. In three dimensions (3D), the integer topological quantum number in a TRI superconductor is determined by the sign of the pairing order parameter and the first Chern number of the Berry phase gauge field on the Fermi surfaces. In two (2D) and one (1D) dimension, the $Z_2$ topological quantum number of a TRI superconductor is determined simply by the sign of the pairing order parameter on the Fermi surfaces. We also obtain a generic and explicit expression of the $Z_2$ topological invariant in 1D and 2D.
Topological superconductors are gapped superconductors with protected Majorana surface/edge states on the boundary. In this paper, we study the Josephson coupling between time-reversal invariant topological superconductors and s-wave superconductors. The Majorana edge/surface states of time-reversal invariant topological superconductors in all physical dimensions 1, 2, 3 have a generic topological property which we named as time-reversal anomaly. Due to the time-reversal anomaly, the Josephson coupling prefers a nonzero phase difference between topological and trivial superconductors. The nont…
We show that under certain conditions multiband superconductors with broken time-reversal symmetry have a new vortex viscosity-generating mechanism which is different from that in conventional superconductors. It appears due to the existence of mixed superfluid phase-density mode inside vortex core. This new contribution is dominant near the time reversal symmetry breaking phase transition. The results could be relevant for three band superconductor $Ba_{1-x}K_{x}Fe_2As_2$.
The topological invariants of gapped time reversal invariant lattice superconductors are studied by mapping the superconducting mean field Hamiltonian to a Bloch Hamiltonian. There is a single $Z_2 $ invariant in two dimensions and four such invariants in three dimensions. We briefly discuss the properties of states with non-trivial topological invariants.
In this paper we study the order parameter and density of states near a grain boundary between two $d_{x^2-y^2}$ superconductors. We examine broken time reversal symmetry near the interface. In particular we show that, under suitable circumstances, time reversal symmetry must be broken even when the order parameter is purely $d_{x^2-y^2}$ everywhere in space.
A novel time-reversal symmetry breaking state is found theoretically in the Josephson junction between the two-gap superconductor and the conventional s-wave superconductor. This occurs due to the frustration between the three order parameters analogous to the two antiferromagnetically coupled XY-spins put under a magnetic field. This leads to the interface states with the energies inside the superconducting gap. Possible experimental observations of this state with broken time-reversal symmetry are discussed.
In superconductors with three or more bands, time-reversal symmetry (TRS) may be broken in the presence of repulsive interband couplings, resulting in a pair of degenerate states characterized by opposite chiralities. We consider a Josephson junction between a three-band superconductor with broken TRS and a single-band superconductor. Phenomena such as asymmetric critical currents, subharmonic Shapiro steps and symmetric Fraunhhofer patterns are revealed theoretically. Existing experimental results are discussed in terms of the present work.
We discuss the problem of broken time reversal symmetry near grain boundaries in a d-wave superconductor based on a Ginzburg-Landau theory. It is shown that such a state can lead to fractional vortices on the grain boundary. Both analytical and numerical results show the structure of this type of state.
In theory of topological classification, the 2D topological superconductors without time reversal symmetry are characterized by Chern numbers. However, in reality, we find the Chern numbers can not reveal the whole properties of the boundary states of the topological superconductors. We figure out some particle-hole symmetry related $\mathbf{Z}_{2}$ invariants, which provide more additional information of the topological superconductors than the Chern numbers provide. With the $\mathbf{Z}_{2}$ invariant, we define weak and strong topological superconductors in 2D systems. Moreover, we explain …
We study the impurity effect in the time reversal symmetry (${\cal T}$) breaking superconductor based on the Bogoliubov-de Gennes (BdG) equations. In ${\cal T}$-violating superconductors, spontaneous currents are induced around the impurity. The current patterns around the impurity reflect the structures of the Cooper pairs. We investigate impurity problem numerically for two kinds of ${\cal T}$ violating superconductors $(p_{x}\pm {\rm i}p_{y}$ and $d+{\rm i}s)$ and investigate the currents around the impurity. We also study the effects of the magnetic impurity in p-wave ($p_{x}\pm {\rm i}p_{…
We present a study of the basic microscopic model of an $s$-wave superconductor with frustrated interband interaction. When frustration is strong, such an interaction gives raise to a $s+is$ state. This is a $s$-wave superconductor that spontaneously breaks time reversal symmetry. We show that, in addition to the known $s+is$ state, there is additional phase, where the system's bulk is a conventional $s$-wave state, but superconducting surface states break time reversal symmetry. Furthermore, we show that $s+is$ superconductors can have spontaneous boundary currents and spontaneous magnetic fi…
We demonstrate that superconductors which break time-reversal symmetry can exhibit thermoelectric properties, which are entirely different from the Ginzburg mechanism. As an example, we show that in the $s+is$ superconducting state there is a reversible contribution to thermally induced supercurrent, whose direction is not invariant under time-reversal operation. Moreover in contrast to Ginzburg's mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. The effect can be used to confirm or rule out the $s+is$ state, which is widely expected to be realized…
We discuss some experimental signatures associated with the topological structures of unconventional superconductor order parameters of form $d_{x^2-y^2}+ix$, where $x=s,p_x \pm p_y$, or $d_{xy}$. Specifically, we study the topological surface states on the $(110)$ and equivalent surfaces of such superconductors which are observable in Andreev tunneling experiments, as well as evaluate the magnetic flux trapped in superconducting rings of such superconductors with multiple grain-boundary Josephson junctions. Previous experiments are examined and several new experiments suggested.
Broken time reversal symmetry (BTRS) in d+id' as well as in d+is superconductors is studied and is shown to yield current carrying surface states. We evaluate the temperature and thickness dependence of the resulting spontaneous magnetization and show a marked difference between weak and strong BTRS. We also derive the Hall conductance which vanishes at zero wavevector q and finite frequency w, however at finite q,w it has an unusual structure. The chirality of the surface states leads to quantum Hall effects for spin and heat transport in d+id' superconductors.
Conventional superconductors are strong diamagnets that through the Meissner effect expel magnetic fields. It would therefore be surprising if a superconducting ground state would support spontaneous magnetics fields. Such time-reversal symmetry broken states have been proposed for the high-temperature superconductors, but their identification remains experimentally controversial. Here we show a route to a low-temperature superconducting state with broken time-reversal symmetry that may accommodate currently conflicting experiments. This state is characterised by an unusual vortex pattern in t…
The Hall voltage sign reversal is consistently explained by the model in which vortices with the superconducting and normal state charge carriers are regarded as three subsystems mutually connected by interactions. The equations of motion for these three subsystems are solved simultaneously and a new formula for the Hall resistivity is obtained. It is shown that it is possible to explain qualitatively experimental data by this model.
The results from muon spin relaxation experiments on the non-centrosymmetric intermetallic superconductor LaNiC$_2$ are reported. We find that the onset of superconductivity coincides with the appearance of spontaneous magnetic fields, implying that in the superconducting state time reversal symmetry is broken. An analysis of the possible pairing symmetries suggests only four triplet states compatible with this observation, all of which are non-unitary. They include the intriguing possibility of triplet pairing with the full point group symmetry of the crystal, which is only possible in a non-…
The discovery of new families of unconventional superconductors is important both experimentally and theoretically, especially if it challenges current models and thinking. By using muon spin relaxation in zero-field, time-reversal symmetry breaking has been observed in Re6Hf. Moreover, the temperature dependence of the superfluid density exhibits s-wave superconductivity with an enhanced electron-phonon coupling. This, coupled with the results from isostructural Re6Zr, shows that the Re6X family are indeed a new and important group of unconventional superconductors.
We report the first experimental observation of the two-node thickness dependence of the critical current in Josephson junctions with a ferromagnetic interlayer. Vanishings of the critical current correspond to transitions into pi-state and back into conventional 0-state. The experimental data allow to extract the superconducting order parameter oscillation period and the pair decay length in the ferromagnet. We develope a theoretical approach based on Usadel equations, which takes into account the spin-flip scattering. Results of numerical calculations are in good agreement with the experimen…
Type II Lifshitz invariant and optically active Higgs mode in time-reversal symmetry broken superconductors
Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院凝聚态理论研究所）；Department of Physics, University of Tokyo（东京大学物理学系）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Trans-Scale Quantum Science Institute, University of Tokyo（东京大学跨尺度量子科学研究所）
We consider a simple model of a multidomain superconductor-ferromagnet-superconductor (SFS) Josephson junction. Sign-alternating magnetization $M$ in domains leads to a spatial modulation of the phase difference $ϕ(x)$. Due to this modulation the Josephson critical current $I_{c}$ may have a different sign depending on the ratio of the magnetic flux in a domain, 4$πMa(2d_{F}),$ to the magnetic flux quantum. This phase modulation, but not a nonmonotonic dependence of the local critical current density $j_{c}$, may be the reason for oscillations of the current $I_{c}$ {\bf \}as a function of the…
Nonreciprocal or nonlinear responses in symmetry-broken systems are powerful probes of emergent properties in quantum materials, including superconductors, magnets, and topological materials. Recently, vortex matter has been recognized as a key ingredient of giant nonlinear responses in superconductors with broken inversion symmetry. However, nonlinear effects have been probed as excess voltage only under broken time-reversal symmetry. In this study, we report intrinsic rectification and nonlinear anomalous Hall effect under time-reversal symmetry in the noncentrosymmetric trigonal superconduc…
Broken time reversal symmetry (BTRS) in d wave superconductors is studied and is shown to yield current carrying surface states. The corresponding spontaneous magnetization is temperature independent near the critical temperature Tc for weak BTRS, in accord with recent data. For strong BTRS and thin films we expect a temperature dependent spontaneous magnetization with a paramagnetic anomaly near Tc. The Hall conductance is found to vanish at zero wavevector q and finite frequency w, however at finite q,w it has an unusual structure.
A topological argument is presented for nodal structures of superconducting states with time-reversal invariance. A generic Hamiltonian which describes a quasiparticle in superconducting states with time-reversal invariance is derived, and it is shown that only line nodes are topologically stable in single-band descriptions of superconductivity. Using the time-reversal symmetry, we introduce a real structure and define topological numbers of line nodes. Stability of line nodes is ensured by conservation of the topological numbers. Line nodes in high-Tc materials, the polar state in p-wave pari…
Using numerical simulation we have studied a magnetization distribution and a process of magnetization reversal in nanoscale magnets placed above a superconductor plane. In order to consider an influence of superconductor on magnetization distribution in the nanomagnet we have used London approximation. We have found that for usual values of London penetration depth the ground state magnetization is mostly unchanged. But at the same time the fields of vortex nucleation and annihilation change significantly: the interval where vortex is stable enlarges on 100-200 Oe for the particle above the s…
We show that superconductors with broken time-reversal symmetry have very specific magnetic and electric responses to inhomogeneous heating. A local heating of such superconductors induces a magnetic field with a profile that is sensitive to the presence of domain walls and crystalline anisotropy of superconducting states. A nonstationary heating process produces an electric field and a charge imbalance in different bands. These effects can be measured and used to distinguish $s+is$ and $s+id$ superconducting states in the candidate materials such as Ba$_{1-x}$K$_x$Fe$_2$As$_2$.
We report point contact Andreev Reflection (PCAR) measurements on a high-quality single crystal of the non-centrosymmetric superconductor Re6Zr. We observe that the PCAR spectra can be fitted by taking two isotropic superconducting gaps with Delta_1 ~ 0.79 meV and Delta_2 ~ 0.22 meV respectively, suggesting that there are at least two bands which contribute to superconductivity. Combined with the observation of time reversal symmetry breaking at the superconducting transition from muon spin relaxation measurements (Phys. Rev. Lett. 112, 107002 (2014)), our results imply an unconventional super…
We study the physical properties of an $s$-wave -- normal metal -- $d$-wave junction in terms of the Andreev bound states in the normal metal layer. The phase dependence of bound states with different orientations leads to superconducting states with broken time reversal symmetry for generic orientations of the $d$-wave superconductor crystal. The occurrence of such a state and the associated spontaneous supercurrent along the junction is analyzed in the framework of Ginzburg-Landau theory and by the solution of the Bogolyubov-de Gennes equations.
Noncentrosymmetric superconductors have sparked significant research interests due to their exciting properties, such as the admixture of spin-singlet and spin-triplet Cooper pairs. Here we report $μ$SR and thermodynamic measurements on the noncentrosymmetric superconductor La7Rh3 which indicate a fully established superconducting gap and spontaneous time-reversal-symmetry breaking at the onset of superconductivity. We show that our results pose severe constraints on any microscopic theory of superconductivity in the system. A symmetry analysis identifies ground states compatible with time-rev…
We solve the Bogoliubov-de Gennes equations self-consistently for d-wave superconductors with [110] surfaces. We find spontaneous breaking of time-reversal symmetry (TRS) caused by the spontaneous occurrence of a complex order parameter with an extended s-wave symmetry along the [110] surfaces. In the TRS-breaking phase, the d-wave order parameter itself becomes complex. We also show that vortex-antivortex pairs can appear along the [110] surfaces of nanoribbons. These pairs are detectable with surface sensitive probes.
It is argued that the $d_{x^2-y^2}$ superconductor exhibits the transition into $d_{x^2-y^2} + id_{xy}$ state in the presence of magnetic impurities in certain conditions. Linear coupling between impurity magnetization and the $d_{xy}$ component drives this transition. Violation of time-reversal symmetry and parity occurs spontaneously via the second order transition. In the ordered phase both impurity magnetization and $d_{xy}$ component of the order parameter develop and are proportional to each other.
We consider different effects that arise when time-reversal symmetry breaking superconductors are subjected to an external magnetic field, thus rendering the superconductor to be in the mixed state. We focus in particular on two time-reversal symmetry breaking order parameters which are believed to be realized in actual materials: $p+ıp'$-wave and $d+ıs$- or $d+ıd'$-wave. The first order parameter is relevant for Sr$_2$RuO$_4$, while the latter order parameters have been suggested to exist near surfaces in some of the high-$T_c$ cuprates. We investigate the interplay between surface states and…
We show that time-reversal invariant superconductors in d=2 (d=3) dimensions can support topologically stable Fermi points (lines), characterized by an integer topological charge. Combining this with the momentum space symmetries present, we prove analogs of the fermion doubling theorem: for d=2 lattice models admitting a spin X electron-hole structure, the number of Fermi points is a multiple of four, while for d=3, Fermi lines come in pairs. We show two implications of our findings for topological superconductors in d=3: first, we relate the bulk topological invariant to a topological number…
We propose a minimal lattice model for two-dimensional class DIII superconductors with $C_2$-protected higher-order topology. While this class of superconductors cannot be topologically characterized by symmetry eigenvalues at high symmetry momenta, we propose a simple Wannier-orbital-based real-space diagnosis to unambiguously capture the corresponding higher-order topology. We further identify and characterize a variety of conventional topological phases in our minimal model, including a weak topological superconductor and a nodal topological superconductor with chiral-symmetry protection. T…
In order to examine the influences of the spatial dependence of the pair potential, the d.c. Josephson current in d-wave superconductors is calculated using self-consistently determined pair potentials. The results show that the suppression of the d-wave pair potential near the insulator does not have serious effect on the properties of the Josephson current. On the other hand, drastic changes are obtained due to the inducement of a subdominant s-wave component, which spontaneously breaks time reversal symmetry. Especially, a rapid enhancement of the Josephson current at low temperature predic…
Recently, a theoretical study [Li {\it et al.},~npj Quantum Materials 6, 36 (2021)] investigated a model of a disordered $d$-wave superconductor, and reported local time-reversal symmetry breaking current loops for sufficiently high disorder levels. Since the pure $d$-wave superconducting state does not break time-reversal symmetry, it is surprising that such persistent currents arise purely from nonmagnetic disorder. Here we perform a detailed theoretical investigation of such disorder-induced orbital currents, and show that the occurrence of the currents can be traced to local extended $s$-w…
At a (110) surface of a $d_{x^2-y-2}$-wave superconductor, superconducting order is strongly suppressed. In such a situation, ordered states that are forbidden in the bulk may arise. This problem is studied for high-$T_C$ cuprate superconductors by treating the $t-J$ model with extended transfer integrals using the Bogoliubov de Gennes method. It is found that a flux phase with staggered currents along the surface, or an antiferromagnetic state can occur near the surface. Stability of the emergent surface states is different from system to system depending on the shapes of their Fermi surfaces…
Effects of impurity scattering on tunneling conductance in dirty normal-metal/insulator/superconductor junctions are studied based on the Kubo formula and the recursive Green function method. The zero-bias conductance peak (ZBCP) is a consequence of the unconventional pairing symmetry in superconductors. The impurity scattering in normal metals suppresses the amplitude of the ZBCP. The degree of the suppression agrees well with results of the quasiclassical Green function theory. When superconductors have $d$+is-wave pairing symmetry, the time-reversal symmetry is broken in superconductors and…
We have investigated the superconducting state of the noncentrosymmetric superconductor Re6Ti (Tc = 6.0 K) using muon-spin rotation/relaxation (muSR) technique. The zero-field muon experiment shows the presence of spontaneous magnetic fields in the superconducting state, indicating time-reversal symmetry breaking (TRSB). However, the low-temperature transverse field muon measurements suggest nodeless s-wave superconductivity. The time reversal symmetry breaking further confirmed in the stoichiometric composition Re24Ti5. These results indicate that the pairing symmetry is not affected by spin-…
In three-band BCS superconductors with repulsive interband interactions, frustration between the bands can lead to an inherently complex gap function, arising out of a phase difference between the bands in the range 0 and π. Since the complex conjugate of this state is also a solution, the ground state is degenerate, and there appears a time-reversal-symmetry-broken state. In this paper we investigate the existence of this state as a function of interband coupling strength and show how a new phase transition appears between the TRSB and conventional BCS states.
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…
The symmetry properties of the order parameter characterize different phases of unconventional superconductors. In the case of the heavy-fermion superconductor UPt$_3$, a key question is whether its multiple superconducting phases preserve or break time-reversal symmetry (TRS). We tested for asymmetry in the phase shift between left and right circularly polarized light reflected from a single crystal of UPt$_3$ at normal incidence, finding that this so-called polar Kerr effect appears only below the lower of the two zero-field superconducting transition temperatures. Our results provide eviden…
A detailed analysis of the symmetric tetracrystal geometry used in phase-sensitive pairing symmetry experiments on high Tc superconductors is carried out for both bulk and surface time-reversal symmetry-breaking states, such as the d+id' and d+is states. The results depend critically on the substrate geometry. In the general case, for the bulk d+id' (or d+is) state, the measured flux quantization should in general not be too different from that obtained in the pure d-wave case, provided |d'| << |d| (or |s| << |d|). However, in one particular high symmetry geometry, the d+id' state gives result…
The cryptoferromagnetic state (the state with intrinsic domain structure) in superconducting ferromagnets subjected to external magnetic field is studied theoretically. Ferromagnetism originates either from electron spin or the intrinsic angular momentum of Cooper pairs (chiral p-wave superconductors like $\mathrm{Sr_2 Ru O_4}$). The phase transitions towards the Meissner and the mixed states are investigated, and the magnetic phase diagrams are obtained. Cryptoferromagnetism, as a form coexistence of superconductivity and ferromagnetism, can be detected by observation of magnetization curves …
We analyze edge currents and edge bands at the surface of a time-reversal symmetry breaking d_{x^2-y^2}+id_{xy} superconductor. We show that the currents have large Friedel oscillations with two interfering frequencies: \sqrt{2}k_F from sub-gap states, and 2 k_F from the continuum. The results are based independently on a self-consistent slave-boson mean field theory for the t-J model on a triangular lattice, and on a T-matrix scattering theory calculation. The shape of the edge-state band, as well as the particular frequency \sqrt{2}k_F of the Friedel oscillations are attributes unique for th…
Symmetry of the order parameter in some triplet superconductors corresponds to doubly-degenerate chiral states. We predict that in a sufficiently small sample this degeneracy can be lifted via macroscopic quantum tunneling. Moreover, if the initial state is prepared correspondingly, 'instanton'-like quantum behavior is possible in a single-domain sample. In this case coherent oscillations of magnetic momentum should take place, as well as the sign alternations of the boundary currents. In particular, spontaneous Hall-effect recently predicted for ruthenates should posses the oscillating voltag…
We study possible applications of high critical temperature nodal superconductors for the search for Majorana bound states in the DIII class. We propose a microscopic analysis of the proximity effect induced by d-wave superconductors on a semiconductor wire with strong spin-orbit coupling. We characterize the induced superconductivity on the wire employing a numerical self-consistent tight-binding Bogoliubov-de Gennes approach, and analytical considerations on the Green's function. The order parameter induced on the wire, the pair correlation function, and the renormalization of the Fermi poin…
Unconventional superconductors have Cooper pairs with lower symmetries than in conventional superconductors. In most unconventional superconductors, the additional symmetry breaking occurs in relation to typical ingredients such as strongly correlated Fermi liquid phases, magnetic fluctuations, or strong spin-orbit coupling in noncentrosymmetric structures. In this article, we show that the time-reversal symmetry breaking in the superconductor LaNiGa$_2$ is enabled by its previously unknown topological electronic band structure. Our single crystal diffraction experiments indicate a nonsymmorph…
We investigate non-Abelian statistics of Majorana Kramers pairs (MKPs) in a network system of one-dimensional time-reversal invariant topological superconductors by using numerical simulations of braiding dynamics, and examine the tolerance against various perturbations which may cause decoherence of MKPs. We, first, consider effects of a magnetic field which breaks time-reversal symmetry. In contrast to a naive expectation, the non-Abelian braiding of MKPs is robust against applied magnetic fields provided that the initial and final states of a braiding process are invariant under the combina…
We investigate the competition between the extended $s_{\pm}$-wave and $d_{x^2-y^2}$-wave pairing order parameters in the iron-based superconductors. Because of the frustrating pairing interactions among the electron and the hole fermi pockets, a time reversal symmetry breaking $s+id$ pairing state could be favored. We analyze this pairing state within the Ginzburg-Landau theory, and explore the experimental consequences. In such a state, spatial inhomogeneity induces supercurrent near a non-magnetic impurity and the corners of a square sample. The resonance mode between the $s_{\pm}$ and $d_{…
Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of ${}^3$He are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology of these time- reversal invariant superconductors manifests itself at the material surface in the form of helical Majorana modes. In this paper, using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. To characterize the effects of strong disorder, we compute the l…
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…
The Josephson-like interband couplings in multi-band superconductivity exhibit degenerate energy minima, which support states with kinks in phase of superconductivity. When the interband couplings in systems of three or more components are frustrated, the time-reversal symmetry (TRS) can be broken, which generates another type of phase kink between the two time-reversal-symmetry breaking (TRSB) pair states. In this work, we focus on these novel states of phase kinks, and investigate their stability, similarity, differences and physical consequences. Main results are summarized as follows: (1) …
Recent observations of two nodeless gaps in superconducting CeCu$_2$Si$_2$ have raised intensive debates as to its exact gap structure of either sign-reversal ($s^{+-}$) or sign-preserving ($s^{++}$) pairing. Here we investigate the quasiparticle interference (QPI) using realistic Fermi surface topology for both weak and strong interband impurity scatterings. Our calculations of the QPI and integrated antisymmetrized local density of states reveal qualitative distinctions between $s^{+-}$ and $s^{++}$ pairing states, which include the intragap impurity resonance and a significant energy-depend…
Observation of Time-Reversal Symmetry Breaking in the Type-I Superconductor YbSb$_2$
Indian Institute of Science Education and Research Bhopal（印度科学教育研究所博帕尔分校）；Indian Institute of Technology, Kanpur（坎普尔印度理工学院）；University of Warwick（华威大学）；STFC Rutherford Appleton Laboratory（英国科学与技术设施委员会鲁瑟福阿普尔顿实验室）
