Fully gapped two-dimensional superconductors coupled to dynamical electromagnetism are known to exhibit topological order. In this work, we develop a unified low-energy description for spin-singlet paired states by deriving topological Chern-Simons field theories for $s$-wave, $d+id$, and chiral higher even-wave superconductors. These theories capture the quantum statistics and fusion rules of Bogoliubov quasiparticles and vortices and incorporate global continuous symmetries - specifically, spin rotation and conservation of magnetic flux - present in all singlet superconductors. For all such systems, we compute the Hall response for these symmetries and investigate the physics at the edge. In particular, the weakly-coupled phase of a chiral $d+id$ chiral state has a spin Hall coefficient $ν_s=2$ and a vanishing Hall response for the magnetic flux symmetry. We argue that the latter is a generic result for two-dimensional superconductors with gapped photons, thereby demonstrating the absence of a spontaneous magnetic field in the ground state of chiral superconductors. It is also shown that the Chern-Simons theories of chiral spin-singlet superconductors derived here fall into Kitaev's 16-fold classification of topological superconductors.
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 argue that a normal Fermi liquid and a singlet, spin gapped Mott insulator cannot be continuously connected, and that some intermediate phase must intrude between them. By explicitly working out a case study where the singlet insulator is stabilized by orbital degeneracy and an inverted Hund's rule coupling, mimicking a Jahn-Teller effect, we find that the intermediate phase is a superconductor.
One of the most important effects of strong electron correlation in high-Tc cuprates is the formation of Zhang-Rice singlets. By fully accounting for the quantum correlation effect of Zhang-Rice singlet formation, we show that a topological spin texture, skyrmion, is created around a Zhang-Rice singlet in the single-hole-doped CuO2 plane. The skyrmion picture provides a natural connection between the antiferromagnetic correlation and the doping concentration x.
We compute the Hall viscosity and conductivity of non-relativistic two-dimensional chiral superconductors, where fermions pair due to a short-range attractive potential, e.g. $p+\mathrm{i}p$ pairing, and interact via a long-range repulsive Coulomb force. For a logarithmic Coulomb potential, the Hall viscosity tensor contains a contribution that is singular at low momentum, which encodes corrections to pressure induced by an external shear strain. Due to this contribution, the Hall viscosity cannot be extracted from the Hall conductivity in spite of Galilean symmetry. For mixed-dimensional chir…
We show how in principle a coherent coupling between two superconductors of opposite parity can be realized in a three-layer oxide heterostructure. Due to strong intraionic spin-orbit coupling in the middle layer, singlet Cooper pairs are converted into triplet ones and vice versa. This results in a large enhancement of the triplet superconductivity, persisting well above the native triplet critical temperature.
We examine the roles of massless Dirac spinon and spin singlet pair excitations on the phase transition in $d-wave$ superconductors. Although the massless spinon excitations in the presence of the spin singlet pair excitations do not alter the nature of the phase transition at $T = 0$, that is, the XY universality class, they are seen to induce an additional attractive interaction potential between vortices, further stabilizing vortex-antivortex pairs at low temperature for lightly doped high $T_c$ samples.
We propose a theoretical model of quasi-one-dimensional superconductors, with attractive electron-electron interactions dominant in the singlet d-wave channel and sub-dominant in the p-wave channel. We discuss, in the mean field approximation, the effect of a magnetic field applied perpendicularly to the direction of the lowest conductivity. The lowest free energy phase corresponds to a singlet d-wave symmetry in low fields, but to a triplet symmetry in high fields. A first order singlet-triplet phase transition is expected at moderate applied fields of a few teslas. We propose to ascribe the …
The Josephson effect between triplet and singlet superconductors is studied. Josephson current can flow between triplet and singlet superconductors due to the spin-orbit coupling in the spin-triplet superconductor but it is finite only when triplet superconductor has $L_z=-S_z=\pm 1$, where $L_z$ and $S_z$ are the perpendicular components of orbital angular momentum and spin angular momentum of the triplet Cooper pairs, respectively. The recently observed temperature and orientational dependence of the critical current through a Josephson junction between UPt$_3$ and Nb is investigated by cons…
We study the Josephson current, I_J, in a junction consisting of two s-wave superconductors that are separated by a ferromagnetic barrier possessing a magnetic and non-magnetic scattering potential, g and Z, respectively. We discuss the general dependence of I_J on g, Z, and the phase difference ϕbetween the two superconductors. Moreover, we compute the critical current, I_c for given g and Z, and show that it possesses two lines of non-analyticity in the (g, Z)-plane. We identify those regions in the (g, Z)-plane where the Josephson current changes sign with increasing temperature without a c…
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…
The basal area (Cu-Cu grid) of the cuprate superconductors not only tends to shrink on hole doping, as expected from single electron quantum chemistry, but exhibits also an electronically incompressible "hump'' around optimum doping n_opt = 0.16. The hump collapses near critical doping n_crit = 0.19. We analyze the origin of the hump in terms of a classical liquid of interacting incompressible particles in a container with antiferromagnetic walls. Oxygen holes interacting with the wall form singlets, protect themselves against other holes by an incompressible "spin fence'', and thus interact a…
We consider a Josephson junction with an arbitrary transmission coefficient $\mathcal{D}$ between a singlet and a triplet superconductor with the latter order parameter characterized by a d-vector of the form ($k_x\hat{y}-k_y\hat{x}$). Various quantities such as the tunnelling current, spin accumulation, and spin current are calculated via the quasiclassical Green's functions. We also present a symmetry argument on the existence of these quantities and their dependencies on the phase difference across the junction. A physical picture is also given in terms of the Andreev states near the juncti…
Realizations of Majorana fermions in solid state materials have attracted great interests recently in connection to topological order and quantum information processing. We propose a novel way to create Majorana fermions in superconductors. We show that an incipient non-collinear magnetic order turns a spin-singlet superconductor with nodes into a topological superconductor with a stable Majorana bound state in the vortex core; at a topologically-stable magnetic point defect; and on the edge. We argue that such an exotic non-Abelian phase can be realized in extended t-J models on the triangula…
The proximity effect from a spin-triplet $p_x$-wave superconductor to a dirty normal-metal has been shown to result in various unusual electromagnetic properties, reflecting a cooperative relation between topologically protected zero-energy quasiparticles and odd-frequency Cooper pairs. However, because of a lack of candidate materials for spin-triplet $p_x$-wave superconductors, observing this effect has been difficult. In this paper, we demonstrate that the anomalous proximity effect, which is essentially equivalent to that of a spin-triplet $p_x$-wave superconductor, can occur in a semicond…
We develop a theory which describes hybrid structures consisting out of superconducting and ferromagnetic parts. We give two examples for applications. First, we consider a hybrid structure containing a strong ferromagnet in the ballistic limit. Second, we study for a weak ferromagnet the influence of a domain wall on the superconducting proximity effect. In both cases we account quantitatively for the mixing between singlet and triplet correlations.
We study the current-phase relation of a ballistic SIFIT junction, consisting of a spin-singlet superconductor (S), a weak ferromagnetic metal (F), a spin-triplet superconductor (T), and insulating ferromagnetic interfaces (I). We use the generalized quasiclassical formalism developed by A. Millis et al. to compute the current density and the free energy of the junction for arbitrary orientation of the magnetizations of the junction barrier. We investigate in detail the effect of the distribution of magnetization on the various harmonics of the current-phase relation and the transition of the …
We analyze general properties of the effective Nambu two-particle vertex and its renormalization group flow in a spin-singlet superconductor. In a fully spin-rotation invariant form the Nambu vertex can be expressed by only three distinct components. Solving exactly the flow of a mean-field model with reduced BCS and forward scattering interactions, we gain insight into the singularities in the momentum and energy dependences of the vertex at and below the critical energy scale for superconductivity. Using a decomposition of the vertex in various interaction channels, we manage to isolate sing…
We study the Josephson effect in spin-singlet superconductor/helical $p$-wave superconductor junctions with a ferromagnetic barrier using the quasiclassical Green function method. It is found that both $\sinϕ$-type and $\cosϕ$-type current-phase relations always exist, irrespective of the gap symmetries in superconductors. The indispensable condition for the $\sinϕ$-type and $\cosϕ$-type current is that the magnetization must have a component parallel to the crystallographic $a$ or $b$ axis, which is distinct from the case of $p$-wave superconductor described by a $\vect{d}$-vector with a unif…
Recently, a strongly enhanced xy anisotropy of magnetic excitations was observed in YBa_2Cu_3O_y (YBCO_y) with y=6.45 and Tc=35 K [Science 319, 597 (2008)]. Unlike the observation in YBCO_6.6 and YBCO_6.85, the anisotropy grows to be pronounced at lower temperature and at lower energy, and is not suppressed by the onset of superconductivity. We propose that the effect of singlet pairing is substantially reduced in YBCO_6.45. This reduction concomitantly enhances an order competing with singlet pairing, a strong tendency of the so-called d-wave Pomeranchuk instability, leading to the magnetic e…
The bulge in the doping dependence of the basal plane area in hole doped cuprate superconductors is connected with a non-double-occupancy constraint for the oxygen cages in the CuO_2 lattice. This constraint favors the formation of 3a hole pairs growing to filaments with gapless excitations along the (pi, pi) direction. Thus in the pseudogap regime a nodal metal of hole pairs is created. Densest packed 3a hole pairs stabilize the optimum doped state at n =1/6 ~ 0.16.
Effects of inhomogeneous doping on the high-$T_c$ cuprate superconductors are studied within the framework of the t-J model. Especially, the boundary between two non-superconducting regions with doping rates much higher and lower than the optimal one is examined. It is found that, although there is no superconductivity in the bulk, a superconducting region appears at the boundary because singlet resonating valence bond order and holon condensation occur simultaneously in this region. The critical temperature of the induced superconductivity can be higher than that of the optimally doped sample…
An unconventional superconductor is distinguished with two types of gap functions: unitary and non-unitary. This core subject has been concentrated on purely spin-triplet or singlet-triplet mixed superconductors. However, the generalization to a purely spin-singlet superconductor has remained primarily of theoretical interest, which requires at least a multi-orbital correlated electronic systems. In this work, we present a possible establishment of both unitary and non-unitary pairings for spin-singlet superconductors with two atomic orbitals. Then we investigate the effects of atomic spin-orb…
We examine the topological properties of a spin-singlet superconductor with Rashba and Dresselhaus (110) spin-orbit couplings. We demonstrate that there are several topological invariants in the Bogoliubov-de Gennes (BdG) Hamiltonian by symmetry analysis. We use the Pfaffian invariant $\mathcal{P}$ for the particle-hole symmetry to demonstrate all the possible phase diagrams of the BdG Hamiltonian. We find that the edge spectrum is either Dirac cone or flat band which supports the emergence of the Majorana fermion in this system. For the Majorana flat bands, a higher symmetric BdG Hamiltonian …
A brief summary of collective mode excitations that can exist in singlet superconductors with irreducible representation $L$ is given. Such excitations may be classified as the coupled excitations of the charge density $ρ$ and the phase $ϕ$ of the order parameter, or of the amplitude $Δ$ of order parameter. Each of these classes may be further characterized in the long wavelength limit by the irreducible representation $\ell$ of the excitation, which may or may not be the same as the ground state $L$.
A theoretical study is presented on the odd-frequency spin-singlet pairing that arises in nonuniform even-frequency superconductors as a consequence of broken translation symmetry. The effect of the odd-frequency pairing on the superfluid density and the spin susceptibility is analyzed by using the quasiclassical theory of superconductivity. It is shown that (1) the superfluid density is reduced by the formation of the odd-frequency singlet pairs and (2) the odd-frequency pairing increases the spin susceptibility even though its spin symmetry is singlet. The two unusual phenomena are related t…
We derive gap equations for superconductivity in coexistence with ferromagnetism. We treat singlet states and triplet states with either equal spin pairing (ESP) or opposite spin pairing (OSP) states, and study the behaviour of these states as a function of exchange splitting. For the s-wave singlet state we find that our gap equations correctly reproduce the Clogston--Chandrasekhar limiting behaviour and the phase diagram of the Baltensperger--Sarma equation (excluding the FFLO region). The singlet superconducting order parameter is shown to be independent of exchange splitting at zero temper…
We investigate the structure of vortex-bound states in spin-singlet chiral superconductors with ($d_{x^2-y^2} \pm i d_{xy}$)-wave and ($d_{xz} \pm id_{yz}$)-wave pairing symmetries. It is found that vortices in the ($d_{xz} \pm id_{yz}$)-wave state bind zero-energy states which are dispersionless along the vortex line, forming a doubly degenerate Majorana flat band. Vortex-bound states of ($d_{x^2-y^2} \pm i d_{xy}$)-wave superconductors, on the other hand, exist only at finite energy. Using exact diagonalization and analytical solutions of tight-binding Bogoliubov-de Gennes Hamiltonians, we c…
Quasi-low-dimensional type II superconductors in parallel magnetic fields are studied when singlet pairing interactions and relatively weak triplet pairing interactions coexist. Singlet and triplet components of order parameter are mixed at high fields, and at the same time an inhomogeneous superconducting state called a Fulde-Ferrell-Larkin-Ovchinnikov state occurs. As a result, the triplet pairing interactions enhance the upper critical field of superconductivity remarkably even at temperatures far above the transition temperature of parallel spin pairing. It is found that the enhancement is…
It is shown that, in a conductor carrying a moderate spin current, singlet pairing can still take place almost as effectively as without a spin current. The system will still be fully gapped for all spin-up and -down single-particle excitations, with no depaired electrons. All thermodynamic properties will be practically the same as without the spin current. All universality relations and laws of corresponding states found in the theory of Bardeen, Cooper, and Schrieffer should remain valid. Thus a singlet-pairing superconductor can always carry a persistent and dissipation-less spin-current. …
In present work the effective singlet-triplet model for CuO2-layer on the grounds of multiband p-d model of strongly correlated electrons is obtained. The resulting Hamiltonian has a form of generalized singlet-triplet t-t'-J model for p-type superconductors and form of usual t-t'-J model for n-type superconductors. In the mean field approximation in X-operator representation we derived equations for Gorkov type Green functions. The symmetry classification of the superconducting order parameter in case of tetragonal lattice resulted in d_{x^2-y^2}- and d_{xy}-types of singlet pairing for both …
We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a super…
We study the Josephson effect in a triplet superconductor--ferromagnet--singlet superconductor junction. We show that the interaction of tunneling Cooper pairs with the interface magnetization can permit a Josephson current at the lowest order of a tunneling Hamiltonian perturbation theory. Two conditions must be satisfied for this to occur: the magnetization of the ferromagnet has a component parallel to the ${\bf d}$-vector of the triplet superconductor, and the gaps of the superconductors have the same parity with respect to the interface momentum. The resulting charge current displays an u…
Dynamical Josephson Effect Between a Singlet and a Triplet Superconductor
Norwegian University of Science and Technology（挪威科技大学）；Imperial College London（伦敦帝国理工学院）；University at Buffalo, State University of New York（纽约州立大学布法罗分校）
Spin splitting, Kondo correlation and singlet-doublet quantum phase transition in a superconductor-coupled InSb nanosheet quantum dot
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Hangzhou International Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）；Peking University（北京大学）
We study topological phases of time-reversal invariant singlet superconductors in three spatial dimensions. In these particle-hole symmetric systems the topological phases are characterized by an even-numbered winding number $ν$. At a two-dimensional (2D) surface the topological properties of this quantum state manifest themselves through the presence of $ν$ flavors of gapless Dirac fermion surface states, which are robust against localization from random impurities. We construct a tight-binding model on the diamond lattice that realizes a topologically nontrivial phase, in which the winding n…
One of the key effects which is predicted to arise in superconductors without a centre of inversion is the mixing of singlet and triplet order parameters, which are no longer good quantum numbers on their own due to parity. We have probed the gap structure in the noncentrosymmetric superconductor Ru$_7$B$_3$, through small-angle neutron diffraction from the vortex lattice, in order to search for the proposed mixed order parameter. We find that the measured temperature dependence of the vortex-lattice form factor is well characterised by a model constructed to describe the effects of broken inv…
We have investigated the proximity effect in superconductor/ferromagnet junctions in a systematic manner to discuss the relationship between the zero-energy peak (ZEP) of the local density of states (LDOS) and spin-triplet odd-frequency pairing. By exactly solving the nonlinear Usadel equations, we have found that the ZEP is realized in a wide range of geometrical and material parameters in the case of the noncollinear magnetization. This strongly suggests the robustness of the ZEP induced by spin-triplet odd-frequency pairing in such systems. We also found that the crossover from singlet pair…
Roles of multipole degrees of freedom in multiband superconductors are investigated in a case of impurities whose low-lying states consist of singlet ground and triplet excited states, which is related to the experimental fact that the transition temperature $T_{\rm c}$ is increased by Pr substitution for La in LaOs$_4$Sb$_{12}$. The most important contribution to the $T_{\rm c}$ increase comes from the inelastic interband scattering of electrons coupled to quadrupole or octupole moments of impurities. It is found that a magnetic field modifies an effective pairing interaction and the scatteri…
Andreev reflection spectroscopy with unpolarized and highly spin-polarized currents has been utilized to study an intermetallic single-crystal superconductor NiBi3. Magnetoresistance at zero bias voltage of point contacts shows the occurrence and suppression of Andreev reflection by unpolarized and polarized current, respectively. The gap value, its symmetry and temperature dependence have been determined using an unpolarized current. The spin state in the NiBi3 sample is determined to be antiparallel using a highly spin-polarized current. The gap value 2Delta/kBT, gap symmetry and its tempera…
Infrared (20-120 cm-1 and 900-1100 cm-1) Faraday rotation and circular dichroism are measured in high Tc superconductors using sensitive polarization modulation techniques. Optimally doped YBCO thin films are studied at temperatures down to 15 K and magnetic fields up to 8 T. At 1000 cm-1 the Hall conductivity varies strongly with temperature in contrast to the longitudinal conductivity which is nearly independent of temperature. The Hall scattering rate has a T^2 temperature dependence but, unlike a Fermi liquid, depends only weakly on frequency. The experiment puts severe constraints on theo…
Presented is the theory of thermal Hall conductivity $κ_{xy}$ in the vortex state of high temperature cuprate superconductors in the clean limit. We show that $\lim_{T \to 0} κ_{xy}/T$ is a {\em staircase} function of 1/B with an envelope that scales as 1/B. The relation to the experiments is discussed.
We calculate the Hall conductivity for a spin-triplet superconductor, using a generalized pairing symmetry dependent on an arbitrary phase, $\vphi$. A promising candidate for such an order parameter is Sr$_{2}$RuO$_{4}$, whose superconducting order parameter symmetry is still subject to investigation. The value of this phase can be determined through Kerr rotation and DC Hall conductivity measurements. Our calculations impose significant constraints on $\vphi$.
We investigate anomalous Hall effect in a magnet coupled to a triplet superconductor under phase gradient. It is found that the anomalous Hall supercurrent arises from non-trivial structure of the magnetization. The magnetic structure manifested in the Hall supercurrent is characterized by even order terms of the exchange coupling, essentially different from that discussed in the context of anomalous Hall effect, reflecting the disspationless nature of supercurrent. We also discuss a possible candidate for magnetic structure to verify our prediction.
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…
Have you been lying awake wondering what symmetries determine whether a superconductor is spin singlet, triplet, or both? We show that if BCS theory is supplied with additional degrees of freedom, spin singlet can coexist with spin triplet superconductivity. In the first part, we didactically guide the reader to the most general superconducting state using symmetry arguments. If both singlet and triplet pairing channels are present, a magnetic field can convert between spin singlet and triplet states. In the second part, we discuss two possible singlet-triplet superconductors: CeRh$_2$As$_2$ a…
We study junctions between superconductors mediated by the edge states of a quantum spin Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a 4π, rather than a 2πperiodicity. This effect is a consequence of the conservation of fermion parity - the number of electrons modulo 2 - in a superconducting junction, and is closely related to the Z_2 topological structure of the quantum spin Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent…
We study the properties of the "spin quantum Hall fluid" - a novel spin phase with quantized spin Hall conductance that is potentially realizable in superconducting systems with unconventional pairing symmetry. A simple realization is provided by a $d_{x^2-y^2} + id_{xy}$ superconductor which we argue has a dimensionless spin Hall conductance equal to two. A theory of the edge states of the $d_{x^2-y^2}+id_{xy}$ superconductor is developed. The properties of the transition to a phase with vanishing spin Hall conductance induced by disorder are considered. We construct a description of this tra…
Magnetotransport theory of layered superconductors in the flux flow steady state is revisited. Longstanding controversies concerning observed Hall sign reversals are resolved. The conductivity separates into a Bardeen-Stephen vortex core contribution, and a Hall conductivity due to moving vortex charge. This charge, which is responsible for Hall anomaly, diverges logarithmically at weak magnetic field. Its values can be extracted from magetoresistivity data by extrapolation of vortex core Hall angle from the normal phase. Hall anomalies in YBCO, BSCCO, and NCCO data are consistent with theoret…
Layered singlet paired superconductors with disorder and broken time reversal symmetry are studied. The phase diagram demonstrates charge-spin separation in transport. In terms of the average intergrain transmission and the interlayer tunnelling we find quantum Hall phases with spin Hall coefficients of 0 and 2 separated by a spin metal phase. We identify a spin metal-insulator localization exponent as well as a spin conductivity exponent of ~0.9. In presence of a Zeeman term an additional phase with spin Hall coefficient of 1 appears.
The long-range proximity effect in superconductor/ferromagnet (S/F) hybrid nano-structures is observed if singlet Cooper pairs from the superconductor are converted into triplet pairs which can diffuse into the fer- romagnet over large distances. It is commonly believed that this happens only in the presence of magnetic inhomogeneities. We show that there are other sources of the long-range triplet component (LRTC) of the con- densate and establish general conditions for their occurrence. As a prototypical example we consider first a system where the exchange field and spin-orbit coupling can …
