The behavior of a dc SQUID, based on a dirty point contacts between a single-band and three-band superconductor with broken time-reversal symmetry is investigated. Using earlier obtained results for Josephson effects in such systems new features in characteristics of a dc SQUID are revealed. It is shown that in the case of a BTRS (broken time-reversal symmetry) three-band superconductor for the applied external magnetic flux, which is divisible by the half-integer flux, strong degeneracy of ground states of a dc SQUID is taken place. This can lead to the appearance of possible multi-hysteresis loops on a dependence of a total flux in the dc SQUID from the externally applied flux. The number of these loops depends on the position of ground states of a three-band superconductor. Also it is found that dependencies of a critical current on applied magnetic flux can have complicated multi-periodic forms, which are differ from strictly periodic characteristics for conventional dc SQUIDs and Fraunhofer patterns for Josephson contacts in the external magnetic field.
Usually, the superconducting quantum interference device (SQUID) consists of two Josephson junctions and the interference therein is modulated by a magnetic flux. In this work, we propose an electrically modulated SQUID consisting of single Josephson junction coupled by a time-reversal breaking Weyl semimetal thin film. For a low Fermi energy, the Josephson current is only mediated by Fermi arc surface states, and has an arbitrary ground-state phase difference \phi0 which is directly proportional to the product of the transverse electric field and the cross section area of the junction.For a s…
We investigate Andreev reflection in two-dimensional heterojunctions formed by a superconductor in contact with a topological insulator ribbon either possessing or breaking time-reversal symmetry. Both classes of topological insulators exhibit perfect Andreev reflection, which is robust against disorder. This is assigned to topologically protected edge states. In the time-reversal symmetric case we show that doping one of the ribbon edges with magnetic impurities suppresses one Andreev channel, while no such suppression is seen in the broken symmetry situation. Based on this observation we sug…
Within a Ginzburg-Landau formalism we establish analytically the necessary and sufficient conditions to realize a doubly degenerate superconducting ground state with broken time-reversal symmetry (BTRS) in a multi-band superconductor. Using these results we analyze the ground state of a three-band superconductor in the cylindrical geometry in an external magnetic field. We show that depending on the interband coupling constants, a magnetic flux can induce current density jumps in such superconducting geometries that are related to adiabatic or non-adiabatic transitions from BTRS to time-revers…
The breaking of time-reversal symmetry in a triplet superconductor Josephson junction is shown to cause a magnetic instability of the tunneling barrier. Using a Ginzburg-Landau analysis of the free energy, we predict that this novel functional behaviour reflects the formation of an exotic Josephson state, distinguished by the existence of fractional flux quanta at the barrier. The crucial role of the orbital pairing state is demonstrated by studying complementary microscopic models of the junction. Signatures of the magnetic instability are found in the critical current of the junction.
We discuss thermal transport of two-dimensional topological superconductors (TSCs) with broken time reversal symmetry, which are described by Bogoliubov-de Gennes (BdG) Hamiltonians. From the calculations of bulk quantities only, without refereeing to Majorana edge states, we show that the thermal Hall conductivity of two-dimensional TSCs in the low-temperature limit is quantized in multiples of $1/2\frac{πT}{6}$, which is exactly one half of the value of quantization in the case of the integer quantum Hall effect, and that this exact half-quantization is caused by the structure of the Nambu s…
We present results of numerical studies of spin quantum Hall transitions in disordered superconductors, in which the pairing order parameter breaks time-reversal symmetry. We focus mainly on p-wave superconductors in which one of the spin components is conserved. The transport properties of the system are studied by numerically diagonalizing pairing Hamiltonians on a lattice, and by calculating the Chern and Thouless numbers of the quasiparticle states. We find that in the presence of disorder, (spin-)current carrying states exist only at discrete critical energies in the thermodynamic limit, …
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.
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 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 $Ba_{1-x}K_{x}Fe_2As_2$.
Spatial distributions of spontaneous magnetic fields near a surface of cuprate high-$T_C$ superconductor YBCO with broken time-reversal symmetry are calculated using the Ginzburg-Landau theory derived from the $t-J$ model. It is found that the magnetic field exists in a narrow region inside the superconductor, and it decays quickly outside the surface. Experimental approaches possible to detect such spontaneous magnetic fields are discussed.
Comment on "High Resolution Polar Effect Measurements of Sr2RuO4: Evidence for Broken Time-Reversal Symmetry in the Superconducting State" by J. Xia et al., Phys. Rev. Lett. 97, 167002 (2006).
Polar Kerr effect in the spin-triplet superconductor Sr2RuO4 was measured with high precision using a Sagnac interferometer with a zero-area Sagnac loop. We observed non-zero Kerr rotations as big as 65 nanorad appearing below Tc in large domains. Our results imply a broken time reversal symmetry state in the superconducting state of Sr2RuO4, similar to He3-A.
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…
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.
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 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.
Recently, Kaminski et al. have reported that time reversal symmetry is broken in the pseudogap phase in the high temperature superconducting material Bi_2Sr_2CaCu_2O_{2+δ} (Bi-2212). Here we examine the role of translationally invariant broken time reversal states on d_{x^2-y^2} superconductors. In particular, we determine the change in the superconducting order parameter structure. We find that the broken time reversal pseudogap state that is consistent with the experiment of Kaminski et al., gives rise to a novel mixed singlet-triplet pairing d+ip phase. This d+ip state is shown to give rise…
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…
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.
Bulk TiSe$_2$ is an intrinsically layered transition metal dichalcogenide (TMD) hosting both superconducting and charge density wave (CDW) ordering. Motivated by the recent progress in preparing two-dimensional TMDs, we study these frustrated orderings in {\it single} trilayer of TiSe$_2$ within a renormalization group approach. We establish that a novel state with time-reversal symmetry broken chiral superconductivity can emerge from the strong competition between CDW formation and superconductivity. Its stability depends on the precise strength and screening of the electron-electron interact…
We demonstrate that films of unconventional (d-wave) superconductor at temperatures $T\lesssim 0.43 T_c$ can exhibit unusual superconducting phases. The new ground states beside the broken gauge and the point group symmetries can spontaneously break (i) continuous translational symmetry and form periodic order parameter structures in the plane of the film, or (ii) time-reversal symmetry and develop supercurrent flowing along the film. These states are result of the strong transverse inhomogeniety present in films with thickness of several coherence lengths. We show a natural similarity between…
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.
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…
Using Monte Carlo simulations, we explore the phase diagram and the phase transitions in $\groupUZ$ $n$-band superconductors with spontaneously broken time-reversal symmetry (also termed $s+is$ superconductors), focusing on the three-band case. In the limit of infinite penetration length, the system under consideration can, for a certain parameter regime, have a single first order phase transition from a $\groupUZ$ broken state to a normal state due to a nontrivial interplay between $\groupU$ vortices and $\groupZ$ domain walls. This regime may also apply to multicomponent superfluids. For oth…
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 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…
Over the last years a lot of theoretical and experimental efforts have been made to find states with broken time reversal symmetry (BTRS) in multi-band superconductors. In particular, it was theoretically proposed that in the Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ system either an $s+is$ or an $s+id$ BTRS state may exist at high doping levels in a narrow region of the phase diagram. Here we report the observation of an enhanced zero field muon spin relaxation rate below the superconducting transition temperature for a high quality crystalline sample with $x \approx$ 0.73. This indicates that indeed the…
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.
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…
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…
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.
We report the results of a search for spontaneous magnetism due to a time reversal symmetry breaking phase in the superconducting state of (110)-oriented YBCO films, expected near the surface in this geometry. Zero field and weak transverse field measurements performed using the low-energy muon spin rotation technique with muons implanted few nm inside optimally-doped YBCO-(110) films show no appearance of spontaneous magnetic fields below the superconducting temperature down to 2.9 K. Our results give an upper limit of ~0.02 mT for putative spontaneous internal fields.
Clusters of interacting two-level-systems (TLS),likely due to $F^+$ centers at the metal-insulator interface, are shown to self consistently lead to $1/f^{α}$ magnetization noise in SQUIDs. By introducing a correlation-function calculation method and without any a priori assumptions on the distribution of fluctuation rates, it is shown why the flux noise is only weakly temperature dependent with $α\lesssim 1$, while the inductance noise has a huge temperature dependence seen in experiment, even though the mechanism producing both spectra is the same. Though both ferromagnetic- RKKY and short-r…
Recently the superconductor and topological semimetal PbTaSe$_2$ was experimentally found to exhibit surface-only lattice rotational symmetry breaking below $T_c$. We exploit the Ginzburg-Landau free energy and propose a microscopic two-channel model to study possible superconducting states on the surface of PbTaSe$_2$. We identify two types of topological superconducting states. One is time-reversal invariant and preserves the lattice hexagonal symmetry while the other breaks both symmetries. We find that such time-reversal symmetry breaking is unavoidable for a superconducting state in a two…
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 study theoretically the spatial distribution of spontaneous magnetic fields near a (110) surface of cuprate high-$T_C$ superconductor YBCO by treating a bilayer $t-J$ model with the Bogoliubov-de Gennes method. Near a (110) surface where $d_{x^2-y^2}$-wave superconductivity is strongly suppressed, flux phase order occurs locally leading to a time-reversal symmetry broken surface state. Since the spontaneous currents flowing in the flux phase in two layers are antiparallele, resulting spontaneous magnetic fields are quite small. They exist in a narrow region inside the superconductor, and es…
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…
Superconductivity is associated with spontaneously broken gauge symmetry. In some exotic superconductors the time-reversal symmetry is broken as well, accompanied with internal magnetic field. A time-reversal symmetry broken (TRSB) superconductivity without internal magnetic field involved can be induced by frustrated inter-component couplings, which becomes a realistic issue recently due to the discovery of iron-pnictide superconductors. Here we derive stability condition for this novel TRSB state using the Ginzburg-Landau (GL) theory. We find that there are multiple divergent coherence lengt…
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.
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…
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…
The noncentrosymmetric superconductor Re$_6$Zr has attracted much interest due to the observation of broken time-reversal symmetry in the superconducting state. Here we report an investigation of the superconducting gap structure of Re$_6$Zr single crystals by measuring the magnetic penetration depth shift $Δλ(T)$ and electronic specific heat $C_e(T)$. $Δλ(T)$ exhibits an exponential temperature dependence behavior for $T~\ll~T_c$, which indicates a fully-open superconducting gap. Our analysis shows that a single gap $s$-wave model is sufficient to describe both the superfluid density $ρ_s(T)$…
We study fermions in a lattice, with on-site and nearest neighbor attractive interactions between two spin species. We consider two geometries: both spins in a triangular lattice, and a mixed geometry with up-spins in honeycomb and down-spins in triangular lattices. We focus on the interplay between spin-population imbalance, on-site and valence bond pairing, and order parameter symmetry. The mixed geometry leads to a rich phase diagram of topologically non-trivial phases. In both geometries, we predict order parameters with simultaneous time-reversal and translational symmetry breaking.
We discuss the phase diagram and phase transitions in $U(1)\times \groupZ{2}$ three-band superconductors with broken time reversal symmetry. We find that beyond mean field approximation and for sufficiently strong frustration of interband interactions there appears an unusual metallic state precursory to a superconducting phase transition. In that state, the system is not superconducting. Nonetheless, it features a spontaneously broken $\groupZ{2}$ time reversal symmetry. By contrast, for weak frustration of interband coupling the energy of a domain wall between different $\groupZ{2}$ states i…
We calculate the tunneling conductance spectra of a ferromagnetic metal/insulator/superconductor using the Blonder-Tinkham-Klapwijk (BTK) formulation. Two possible states for the superconductor are considered with the time reversal symmetry ($\cal{T}$) broken, i.e., $d_{x^2-y^2}+is$, or $d_{x^2-y^2}+id_{xy}$. In both cases the tunneling conductance within the gap is suppressed with the increase of the exchange interaction due to the suppression of the Andreev reflection. In the $(d_{x^2-y^2}+is)$-wave case the peaks that exist when the ferromagnet is a normal metal in the amplitude of the s-wa…
Weak spontaneous magnetic fields are observed near the surface of YBCO films using Beta-detected Nuclear Magnetic Resonance. Below Tc, the magnetic field distribution in a silver film evaporated onto the superconductor shows additional line broadening, indicating the appearance of small disordered magnetic fields. The line broadening increases linearly with a weak external magnetic field applied parallel to the surface, and is depth-independent up to 45 nm from the Ag/YBCO interface. The magnitude of the line broadening at 10 K extrapolated to zero applied field is less than 0.2 G, and is clos…
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…
It is commonly believed that in the absence of disorder or an external magnetic field, there are three possible types of superconducting excitation gaps: the gap is nodeless, it has point nodes, or it has line nodes. Here, we show that for an even-parity nodal superconducting state which spontaneously breaks time-reversal symmetry, the low-energy excitation spectrum generally does not belong to any of these categories, instead it has extended Bogoliubov Fermi surfaces. These Fermi surfaces can be visualized as two-dimensional surfaces generated by "inflating" point or line nodes into spheroids…
Recent experimental and theoretical interest in the superconducting phase of the heavy fermion material URu$_2$Si$_2$ has led to a number of proposals in which the superconducting order parameter breaks time-reversal symmetry (TRS). In this study we measured polar Kerr effect (PKE) as a function of temperature for several high-quality single crystals of URu$_2$Si$_2$. We find an onset of PKE below the superconducting transition that is consistent with a TRS-breaking order parameter. This effect appears to be independent of an additional, possibly extrinsic, PKE generated above the hidden order…
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…
