Superconductivity and magnetism are antagonistic states of matter. The presence of spontaneous magnetic fields inside the superconducting state is, therefore, an intriguing phenomenon prompting extensive experimental and theoretical research. In this review, we discuss recent experimental discoveries of unconventional superconductors which spontaneously break time-reversal symmetry and theoretical efforts in understanding their properties. We discuss the main experimental probes and give an extensive account of theoretical approaches to understand the order parameter symmetries and the corresponding pairing mechanisms including the importance of multiple bands.
Paramagnetic effects are shown to result in the appearance of a triplet component of order parameter in a vortex phase of a d-wave superconductor in the absence of impurities. This component, which breaks both parity and time-reversal symmetries of Cooper pairs, is expected to be of the order of unity in a number of modern superconductors such as organic, high-Tc, and some others. A generic phase diagram of such type-IV superconductors, which are singlet ones at H=0 and characterized by singlet-triplet mixed Copper pairs with broken time-reversal symmetry in a vortex phase, is discussed.
A Josephson junction consisting of an Anderson impurity weakly coupled with two-band and single-band superconductors exposes a time reversal breaking ground state when the coupling between the two bands exceeds a certain threshold. The critical regime occurs around local moment formation. This indicates a fundamental and distinct role of strong correlations: Driving a system into a time reversal breaking ground state. One of the observable consequences is that the impurity magnetization in this phase is reduced.
Vortex phase in a singlet superconductor in the absence of impurities is shown to be absolutely unstable with respect to the appearance of a triplet component which breaks both inversion and time-reversal symmetries of Cooper pairs. Symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, Hc2(0), is of the order of Clogston paramagnetic limiting field, Hp. We suggest a generic phase diagram of such type-IV superconductor, which is singlet one at H=0 and characterized by mixed singlet-triplet order parameter with broken time-revers…
We propose a modified Josephson corner-junction experiment which can test whether the order parameter in the iron pnictides changes sign between the electron and hole pockets of the Fermi surface.
We argue that that in their recent experiment in which they claim to have found evidence for a time-reversal symmetry broken state, Kaminski et al. overlooked small temperature dependent changes in the superstructure of Bi2212. These subtle changes may manifest themselves by changing the final state configurations of the photoemission process and thus invalidate their ultimate conclusions.
We show that singlet superconductivity in the Abrikosov vortex phase is absolutely unstable with respect to the appearance of a chiral triplet component of a superconducting order parameter. This chiral component, p_x + i p_y, breaks time-reversal, parity, and spin rotational symmetries of the internal order parameter, responsible for a relative motion of two electrons in the Cooper pair. We demonstrate that the symmetry breaking Pauli paramagnetic effects can be tuned by a magnetic field strength and direction and can be made of the order of unity in organic and high-temperature layered super…
We study the behavior of spinless fermions in superconducting state, in which the phases of the superconducting order parameter depend on the direction of the link. We find that the energy of the superconductor depends on the phase differences of the superconducting order parameter. The solutions for the phases corresponding to the energy minimuma, lead to a topological superconducting state with the nontrivial Chern numbers. We focus our quantitative analysis on the properties of topological states of superconductors with different crystalline symmetry and show that the phase transition in th…
The search for broken time reversal symmetry (TRSB) in unconventional superconductors intensified in the past year as more systems have been predicted to possess such a state. Following our pioneering study of TRSB states in Sr$_2$RuO$_4$ using magneto-optic probes, we embarked on a systematic study of several other of these candidate systems. The primary instrument for our studies is the Sagnac magneto-optic interferometer, which we recently developed. This instrument can measure magneto-optic Faraday or Kerr effects with an unprecedented sensitivity of 10 nanoradians at temperatures as low a…
We show that an isolated impurity in a spin singlet d_{x2-y2} superconductor generates a d_{xy} order parameter with locally broken time-reversal symmetry. The origin of this effect is a coupling between the d_{x2-y2} and the d_{xy} order parameter induced by spin-orbit scattering off the impurity. The signature of locally broken time-reversal symmetry is an induced orbital charge current near the impurity, which generates a localized magnetic field in the vicinity of the impurity. We present a microscopic theory for the impurity induced d_{xy} component, discuss its spatial structure as well …
Surface states of d_{x^2-y^2}-wave superconductors are studied using the Ginzburg-Landau (GL) theory. For a [110] surface it has been known that the time-reversal symmetry (T) breaking surface state, (d+-is)-wave state, can occur if the bare transition temperature of the s-wave order parameter (OP) is positive. We show that even if this bare T_c is negative, it is possible to break T because the coupling to the spontaneously generated magnetic field may induce the s-wave OP. The T-breaking state is favored when the GL parameter (kappa) is small.
In a recent Letter [PRL 103, 186402 (2009)], the energy band structure of Fe-As-based superconductors is fitted with a tight-binding model with two Fe ions per unit cell and two degenerate xz and yz orbitals per Fe ion. The author claims that the proposed model, which differs markedly from a model previously used by other authors for the same two orbitals in the same compounds, possesses the symmetry required to describe the Fe-As planes in iron-pnictide superconductors. In this comment we argue that this is not the case.
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 present a group theoretic analysis of several classes of organic superconductors. We argue that highly frustrated half-filled layered organic superconductors, such as kappa-(ET)_2Cu2(CN)_3 (where ET is BEDT-TTF) and beta'-[Pd(dmit)_2]_2X, undergo two superconducting phase transitions, the first from the normal state to a d-wave superconductor and the second to a d+id state. We show that the monoclinic distortion of kappa-(ET)_2Cu(NCS)_2 means that the symmetry of its superconducting order parameter is different from that of orthorhombic kappa-(ET)_2Cu[N(CN)_2]Br. We propose that quarter fil…
The robustness against local perturbations, as long as the symmetry of the system is preserved, is a distinctive feature of topological quantum states. Magnetic impurities and defects break time-reversal invariance and, consequently, time-reversal invariant (TRI) topological superconductors are fragile against this type of disorder. Non-magnetic impurities, however, preserve time-reversal symmetry and one naively expects a TRI topological superconductor to persist in the presence of non-magnetic impurities. In this work, we study the effect of non-magnetic disorder on a TRI topological superco…
The multipocket Fermi surfaces of iron-based superconductors promote pairing states with both s_{+-}-wave and d_{x^2-y^2}-wave symmetry. We argue that the competition between these two order parameters could lead to a time-reversal-symmetry breaking state with s+id-pairing symmetry in the iron-based superconductors, and propose serveral scenarios in which this phase may be found. To understand the emergence of such a pairing state on a more rigorous footing, we start from a microscopic 5-orbital description representative for the pnictides. Using a combined approach of functional renormalizati…
I study the effects of disorder on the superconductivity of a three-band model with repulsive interband pairing. Such a model can support several possible superconducting order parameters, including a complex time-reversal symmetry breaking (TRSB) state. Impurity scattering suppresses the critical temperature of all these states, but the complex state survives, and remains a part of the phase diagram of the model even in the presence of moderate amount of disorder. This means that the TRSB states could be experimentally accessible in multiband materials like iron pnictides and chalcogenides.
We investigate the superconducting gap function of topological superconductor PbTaSe$_2$. Temperature, magnetic field, and three-dimensional (3D) field-angle dependences of the specific heat prove that the superconductivity of PbTaSe$_2$ is fully-gapped, with two isotropic $s$-wave gaps. The pair-breaking effect is probed by systematically increasing non-magnetic disorders through H$^+$-irradiations. The superconducting transition temperature, $T_{\rm{c}}$, is found to be robust against disorders, which suggests that the pairing should be sign-preserved rather than sign-reversed.
We report the discovery of Zr$_3$Ir as a new type of unconventional noncentrosymmetric superconductor (with $T_c = 2.3$ K), here investigated mostly via muon-spin rotation/relaxation ($μ$SR) techniques. Its superconductivity was characterized using magnetic susceptibility, electrical resistivity, and heat capacity measurements. The low-temperature superfluid density, determined via transverse-field $μ$SR and electronic specific heat, suggests a fully-gapped superconducting state. The spontaneous magnetic fields, revealed by zero-field $μ$SR below $T_c$, indicate the breaking of time-reversal s…
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.
Time-reversal-invariant topological superconductor (TRITOPS) wires host Majorana Kramers pairs that have been predicted to mediate a fractional Josephson effect with $4π$ periodicity in the superconducting phase difference. We explore the TRITOPS fractional Josephson effect in the presence of time-dependent `local mixing' perturbations that instantaneously preserve time-reversal symmetry. Specifically, we show that just as such couplings render braiding of Majorana Kramers pairs non-universal, the Josephson current becomes either aperiodic or $2π$-periodic (depending on conditions that we quan…
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.
We propose a setup to realize time-reversal invariant topological superconductors in quantum wires, proximity coupled to conventional superconductors. We consider a model of quantum wire with strong spin-orbit coupling and proximity coupling to two s-wave superconductors. When the relative phase between the two superconductors is $ϕ=π$ a Kramers' pair of Majorana zero modes appears at each edge of the wire. We study the robustness of the phase in presence of both time-reversal invariant and time-reversal breaking perturbations. In addition, we show that the system forms a natural realization o…
By employing a series of experimental techniques, we provide clear evidence that CaPtAs represents a rare example of a noncentrosymmetric superconductor which simultaneously exhibits nodes in the superconducting gap and broken time-reversal symmetry (TRS) in its superconducting state (below $T_c$ $\approx$ 1.5 K). Unlike in fully-gapped superconductors, the magnetic penetration depth $λ(T)$ does not saturate at low temperatures, but instead it shows a $T^2$-dependence, characteristic of gap nodes. Both the superfluid density and the electronic specific heat are best described by a two-gap mode…
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 …
A novel superconducting state under the broken time-reversal symmetry is studied in conventional phonon-mediated superconductors. By solving the Eliashberg equation self-consistently with the mass renormalization effect, it is found that the even- and odd-frequency components of the order parameter coexist in the bulk system as a consequence of the broken time-reversal symmetry. This finding would direct more attention to the odd-frequency pairing that affects physical quantities, especially in strong coupling superconductors.
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…
In a recent comment [1], Armitage and Hu have suggested that our experiment observing dichroism in angle resolved photoemission (ARPES) [2] could not be conclusively interpreted as arising from time reversal symmetry breaking, arguing that our observations are likely due to structural effects. The concerns expressed by Armitage and Hu that our results could be due to a change in the mirror plane are as important as they are obvious. In fact the first part of their comment merely restates the results of Simon and Varma [3] about the relationship and contrast of effects due to time reversal symm…
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…
To identify the superconducting gap symmetry in CeCoIn5 (Tc=2.3 K), we performed angle-resolved specific heat (C_ϕ) measurements in a field rotated around the c-axis down to very low temperatures 0.05Tc and detailed theoretical calculations. In a field of 1 T, a sign reversal of the fourfold angular oscillation in C_ϕhas been observed at T ~ 0.1Tc on entering a quasiclassical regime where the maximum of C_ϕcorresponds to the antinodal direction, coinciding with the angle-resolved density of states (ADOS) calculation. The C_ϕbehavior, which exhibits minima along [110] directions, unambiguously …
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.
We show that three-band superconductors with broken time reversal symmetry allow magnetic flux-carrying stable topological solitons which have only a slightly higher energy than ordinary vortices. Therefore they can be induced by fluctuations or quenching the system through a phase transition. It can provide an experimental signature of the time reversal symmetry breakdown.
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-…
We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes. The multi-band character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulat…
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…
