Much excitement surrounds the possibility that strontium ruthenate exhibits chiral p-wave superconducting order. Such order would be a solid state analogue of the A phase of He-3, with the potential for exotic physics relevant to quantum computing. We take a critical look at the evidence for such time-reversal symmetry breaking order. The possible superconducting order parameter symmetries and the evidence for and against chiral p-wave order are reviewed, with an emphasis on the most recent theoretical predictions and experimental observations. In particular, attempts to reconcile experimental observations and theoretical predictions for the spontaneous supercurrents expected at sample edges and domain walls of a chiral p-wave superconductor and for the polar Kerr effect, a key signature of broken time-reversal symmetry, are discussed.
We identify an intrinsic Hall effect in multiband chiral superconductors in the absence of a magnetic field (i.e., an anomalous Hall effect). This effect arises from interband transitions involving time-reversal symmetry-breaking chiral Cooper pairs. We discuss the implications of this effect for the putative chiral p-wave superconductor, Sr2RuO4, and show that it can contribute significantly to Kerr rotation experiments. Since the magnitude of the effect depends on the structure of the order parameter across the bands, this result may be used to distinguish between different models proposed f…
A d-wave superconductor, its phase coherence progressively destroyed by unbinding of vortex-antivortex pairs, suffers an instability related to chiral symmetry breaking in two-flavor QED$_3$. The chiral manifold exhibits large degeneracy spanned by physical states acting as inherent ``competitors'' of d-wave superconductivity. Two of these states are associated with antiferromagnetic insulator and ``stripe'' phases, known to be stable in the pseudogap regime of cuprates near half-filling. The theory also predicts additional, yet unobserved state: a d+ip phase-incoherent superconductor.
We find chiral charge order in the superconductor 2H-TaS2 using Scanning Tunneling Microscopy and Spectroscopy (STM/S) at 0.1 K. Topographic images show hexagonal atomic lattice and charge density wave (CDW) with clockwise and counterclockwise charge modulations. Tunneling spectroscopy reveals the superconducting density of states, disappearing at Tc = 1.75 K and showing a wide distribution of values of the superconducting gap, centered around Δ=0.28 meV.
Van der Waals materials like NbSe$_2$ or TaS$_2$ have demonstrated Ising superconductivity down to atomically thin layers. Due to the spin-orbit coupling, these superconductors have the in-plane upper critical magnetic field far beyond the Pauli limit. We theoretically demonstrate that, twisted bilayer Ising superconductors separated by a ferromagnetic buffer layer can naturally host chiral topological superconductivity with Chern numbers, which can be realized in heterostructures like $\mathrm{NbSe_2/CrCl_3/NbSe_2}$. Under appropriate experimental conditions the topological superconducting ga…
The local density of states (LDOS) around a vortex core is studied theoretically in terms of the odd-frequency (odd-omega) Cooper pairing induced around the center of the vortex core. We find a robust zero-energy peak in the LDOS at a vortex core in a chiral p-wave superconductor originating from an odd-omega s-wave pair amplitude. We suggest how to discriminate a spin-triplet pairing symmetry and spatial chiral-domain structure by scanning tunneling spectroscopy via odd-omega pair amplitudes inside vortex cores.
We study surface zero-energy flat bands in three-dimensional chiral superconductors with $p_z (p_x +ip_y)^ν$-wave pairing symmetry ($ν$ is a nonzero integer), based on topological numbers and tunneling conductance. It is shown that the surface flat bands are fragile against (i) the misorientation angle and (ii) surface Rashba spin-orbit interaction. The fragility of (i) is specific to chiral SCs, whereas that of (ii) happens for general odd-parity SCs. We demonstrate that these flat band instabilities vanish or suppress a zero-bias conductance peak in a normal/insulator/superconductor junction…
We study the effects of the leading irrelevant perturbations on chiral symmetry breaking in the effective $QED_3$ theory of d-wave superconductor-insulator transition in underdoped cuprates. For weak symmetry breaking, the effect of a perturbation on energies of various insulating states can be classified according to its engineering dimension in the maximally symmetric theory. Considering the velocity anisotropy, repulsive interactions, and higher order derivatives we show that the insulating state with the lowest energy is the spin density wave.
We investigate theoretically the equilibrium configurations of many magnetic skyrmions interacting with many superconducting vortices in a superconductor-chiral magnet bilayer. We show that miscible mixtures of vortices and skyrmions in this system breaks down at a particular wave number for sufficiently strong coupling, giving place to remarkably diverse mesoscale patterns: gel, stripes, clusters, intercalated stripes and composite gel-cluster structures. We also demonstrate that, by appropriate choice of parameters, one can thermally tune between the homogeneous and density-modulated phases.
We identify an intrinsic mechanism of the anomalous Hall effect for non-symmorphic chiral superconductors. This mechanism relies on both a nontrivial multi-band chiral superconducting order parameter, which is a mixture of pairings of even and odd angular momentum channels, and a complex normal state inter-sublattice hopping, both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply this mechanism to the putative chiral superconducting phase of the heavy-fermion superconductor $\mathrm{UPt_3}$ and calculate the anomalous ac Hall conductivity in a si…
We investigate a hybrid heterostructure with magnetic skyrmions (Sk) inside a chiral ferromagnet interfaced by a thin superconducting film via an insulating barrier. The barrier prevents the electronic transport between the superconductor and the chiral magnet, such that the coupling can only occur through the magnetic fields generated by these materials. We find that Pearl vortices (PV) are generated spontaneously in the superconductor within the skyrmion radius, while anti-Pearl vortices (\(\overline{\mathrm{PV}}\)) compensating the magnetic moment of the Pearl vortices are generated outside…
This paper examines thermodynamic stability of chiral domain walls and vortices - topological defects which can exist in chiral superconductors. Using London theory it is demonstrated that at sufficiently small applied and chiral fields the existence of domain walls and vortices in the sample is not favored and the sample's configuration is a single domain. The particular chirality of the single-domain configuration is neither favored nor disfavored by the applied field. Increasing the field leads to an entry of a domain wall loop or a vortex into the sample. Formation of a straight domain wal…
Quasiparticle states around a single vortex in a $p_x\pm i p_y$-wave superconductor are studied on the basis of the Bogoliubov-de Gennes (BdG) theory, where both charge and current screenings are taken into account. Due to the violation of time reversal symmetry, there are two types of vortices which are distinguished by their winding orientations relative to the angular momentum of the chiral Cooper pair. The BdG solution shows that the charges of the two types of vortices are quite different, reflecting the rotating Cooper pair of the $p_x\pm i p_y$-wave paring state.
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…
We study the local density of states at the surface of a chiral p-wave superconductor in the presence of a weak magnetic field. As a result, the formation of low-energy Andreev bound states is either suppressed or enhanced by an applied magnetic field, depending on its orientation with respect to the chirality of the p-wave superconductor. Similarly, an Abrikosov vortex, which is situated not too far from the surface, leads to a zero-energy peak of the density of states, if its chirality is the same as that of the superconductor, and to a gap structure for the opposite case. We explain the und…
We show that the Majorana fermion zero modes in the cores of odd winding number vortices of a 2D $p_x+ip_y$-paired superconductor is due to an index theorem. This theorem is analogous to that proven by Jackiw and Rebbi for the existence of localized Dirac fermion zero modes on the mass domain walls of a 1D Dirac theory. The important difference is that, in our case, the theorem is proven for a two component fermion theory where the first and second components are related by parity reversal and hermitian conjugation.
Motivated by the recent theoretical and experimental progress in the heavy fermion system UCoGe, we study ferromagnetic chiral superconductors in the presence of magnetic domains. Within mean field approximations, it is shown that chiral superconducting domains are naturally induced by the ferromagnetic domains. The domain wall current flows in the opposite direction to the naively expected one as in $^3$He-A phase due to contributions from "unpaired electrons". Consequently, the domain wall current flows in the same direction with that of surface currents when the magnetic domain wall lies pa…
Motivated by the recent controversy surrounding the Kerr effect measurements in strontium ruthenate \cite{xia:167002}, we examine the electromagnetic response of a clean chiral p-wave superconductor. When the contributions of the collective modes are accounted for, the Hall response in a clean chiral superconductor is smaller by several orders of magnitude than previous theoretical predictions and is too small to explain the experiment. We also uncover some unusual features of the collective modes of a chiral superconductor, namely, that they are not purely longitudinal and couple to external …
We study effects of a chiral helimagnet (CHM) on vortex states in a superconductor, solving the Ginzburg-Landau equations in a chiral helimagnet/superconductor bilayer system. We found that vortices form a periodically modulated triangular lattice, because the magnetic field from the chiral helimagnet $H_{CHM}$ oscillates spatially. An increase of a critical current is expected, because vortices are pinned by the $H_{CHM}$.
We study theoretically the spontaneous edge current in a small chiral superconductor with surface roughness. We obtained self-consistent solutions of the pair potential and the vector potential by solving the quasiclassical Eilenberger equation and the Maxwell equation simultaneously. We then employed them to calculate numerically the spatial distribution of the chiral edge current in a small superconductor. The characteristic behavior of the spontaneous edge current depends strongly on the symmetries of the order parameters such as chiral $p$-, chiral $d$- and chiral $f$-wave pairing. The edg…
We study the pi phase in a superconductor-ferromagnet-superconductor Josephson junction, with a ferromagnet showing a cycloidal spiral spin modulation with in-plane propagation vector. Our results reveal a high sensitivity of the junction to the spiral order and indicate the presence of 0-pi quantum phase transitions as function of the spiral wave vector. We find that the chiral magnetic order introduces chiral superconducting triplet pairs that strongly influence the physics in such Josephson junctions, with potential applications in nanoelectronics and spintronics.
Clapping modes, which are relative amplitude and phase modes between two chiral components of Cooper pairs, are bosonic collective modes inherent to chiral superconductors. These modes behave as long-lived bosons with masses smaller than the threshold energy, $2|Δ|$, for decay into unbound fermion pairs. Here, we clarify that the real/imaginary clapping modes in chiral superconductors directly couple to acoustic wave propagation when the weak particle-hole asymmetry of the normal state quasiparticle dispersion is taken into account. The clapping modes driven by an acoustic wave generate an alt…
The edge currents of two dimensional topological chiral superconductors with nonzero Cooper pair angular momentum---e.g., chiral $p$-, $d$-, and $f$-wave superconductivity---are studied. Bogoliubov-de Gennes and Ginzburg--Landau calculations are used to show that in the continuum limit, \emph{only} chiral $p$-wave states have a nonzero edge current. Outside this limit, when lattice effects become important, edge currents in non-$p$-wave superconductors are comparatively smaller, but can be nonzero. Using Ginzburg--Landau theory, a simple criterion is derived for when edge currents vanish for n…
This is a short note presenting a repeated experiment of the half-quantized transport signature given by Majorana chiral modes living in a heterostructure formed by an s-wave superconductor and a quantum-anomalous-Hall insulator.
We have investigated vortex states in chiral helimagnet/superconductor bilayer systems under an applied external magnetic field Happl, using the Ginzburg-Landau equations. Effect of the chiral helimagnet on the superconductor is taken as a magnetic field HCHM, which is perpendicular to the superconductor and oscillates spatially. For Happl = 0 and weak HCHM, there appear pairs of up-and down-vortices. Increasing Happl, down-vortices gradually disappear, and number of up-vortices increases in the large magnetic field region. Then, up-vortices form parallel, triangular, or square structures.
We demonstrate that two-dimensional chiral superconductors on curved surfaces spontaneously develop magnetic flux. This geometric Meissner effect provides an unequivocal signature of chiral super- conductivity, which could be observed in layered materials under stress. We also employ the effect to explain some puzzling questions related to the location of zero-energy Majorana modes.
Recently, a quantum anomalous Hall insulator (QAHI)/superconductor heterostructure has been realized and shows half-quantized conductance plateaus in two-terminal conductance measurements [Q. L. He \textit{et al.}, Science {\bf357}, 294 (2017)]. The half-quantized conductance plateaus are considered as a solid evidence of chiral Majorana edge modes. However, there is a strong debate over the origin of the half-quantized conductance plateaus. In this work, we propose a Josephson junction based on the QAHI/superconductor heterostructure to identify the existence of chiral Majorana edge modes. We…
Microwave Kerr/Faraday Resonance in Two-dimensional Chiral Superconductors
The University of Osaka（大阪大学）；Institute for Open and Transdisciplinary Research Initiatives, The University of Osaka（大阪大学开放跨学科研究倡议研究所）；Japan Atomic Energy Agency（日本原子能机构）；The University of Tokyo（东京大学）；Louisiana State University（路易斯安那州立大学）；Kyoto University（京都大学）
The vortex core in chiral p-wave superconductors exhibits various properties owing to the interplay between the vorticity and chirality inside the vortex core. In the chiral p-wave superconductors, the site-selective nuclear spin-lattice relaxation rate T1^{-1} is theoretically studied inside the vortex core within the framework of the quasiclassical theory of superconductivity. T1^{-1} at the vortex center depends on the sense of the chirality relative to the sense of the magnetic field. The effect of a tilt of the magnetic field upon T1^{-1} is investigated. The effect of the anisotropy in t…
Chiral superconductivity is a striking quantum phenomenon in which an unconventional superconductor spontaneously develops an angular momentum and lowers its free energy by eliminating nodes in the gap. It is a topologically non-trivial state and, as such, exhibits distinctive topological modes at surfaces and defects. In this paper we discuss the current theory and experimental results on chiral superconductors, focusing on two of the best-studied systems, Sr2RuO4, which is thought to be a chiral triplet p-wave superconductor, and UPt3, which has two low-temperature superconducting phases (in…
We studied strong-coupling effect upon an isolated vortex in a two-dimensional chiral p-wave superconductor. We solved the Eilenberger equation for the quasiclassical Green's functions and the Eliashberg equation with single mode Einstein boson self-consistently. We calculated the free-energy of each obtained vortex, and found that a non-axisymmetric vortex metastably exists in some situation.
We study the Josephson effect between chiral p-wave superconductor / diffusive normal metal (DN) / chiral p-wave superconductor (CP/DN/CP) junctions using quasiclassical Green's function formalism with proper boundary conditions. The px+ipy-wave symmetry of superconducting order parameter is chosen which is believed to be a pairing state in Sr2RuO4. It is shown that the Cooper pairs induced in DN have an odd-frequency spin-triplet s-wave symmetry, where pair amplitude is an odd function of Matsubara frequency. Despite the peculiar symmetry properties of the Cooper pairs, the behavior of the Jo…
In this work we consider the influence of potential impurities deposited on top of two-dimensional chiral superconductors. As discovered recently, magnetic impurity lattices on an $s$-wave superconductor may give rise to a rich topological phase diagram. We show that similar mechanism takes place in chiral superconductors decorated by non-magnetic impurities, thus avoiding the delicate issue of magnetic ordering of adatoms. We illustrate the method by presenting the theory of potential impurity lattices embedded on chiral $p$-wave superconductors. While a prerequisite for the topological state…
We calculate microscopically the interaction energy of domain walls separating degenerate ground states in a chiral p-wave superconductor. The interaction is mediated by the quasiparticles experiencing Andreev scattering at the domain walls. As a by-product, we derive a useful general expression for the free energy of an arbitrary nonuniform texture of the order parameter in terms of the quasiparticle scattering matrix.
The elementary vortex pinning potential is studied in a chiral p-wave superconductor with a pairing d=z(k_x + i k_y) on the basis of the quasiclassical theory of superconductivity. An analytical investigation and numerical results are presented to show that the vortex pinning potential is dependent on whether the vorticity and chirality are parallel or antiparallel. Mutual cancellation of the vorticity and chirality around a vortex is physically crucial to the effect of the pinning center inside the vortex core.
This paper has been withdrawn by the authors. Our basic argument on the gauge invariance presented in this paper cannot withstand the criticism which has been drawn to our attention by several colleagues.
Chiral superconductors are a novel class of unconventional superconductors that host topologically protected chiral Majorana fermions at interfaces and domain walls, elusive quasiparticles that could serve as a platform for topological quantum computing. Here we show that, in analogy to a qubit, the out-of-equilibrium superconducting state in such materials can be described by a Bloch vector and controlled on ultrafast time scales. The all-optical control mechanism is universal, permitting arbitrary rotations of the order parameter, and can induce a dynamical change of handedness of the conden…
We investigate several important properties of multi-band superconductors. They are time-reversal symmetry breaking, chirality and fractional quantum flux vortices in three-band superconductors. The BCS (Bardeen-Cooper-Schrieffer) gap equation has a solution with time-reversal symmetry breaking in some cases. We derive the Ginzburg-Landau free energy from the BCS microscopic theory. The frustrating pairing interaction among Fermi surfaces leads to a state with broken time-reversal symmetry, that is, a chiral solution. The Ginzburg-Landau equation for three-component superconductors leads to a …
We study the tunneling conductance of a clean normal metal/chiral p-wave superconductor junction using the extended Blonder-Tinkham-Klapwijk formalism. It is shown that the spin-orbit coupling of the Rashba type that is present near the interface causes the subgap conductance peaks associated with the Andreev surface bound states to shift to a nonzero bias. We also investigate the effect of the Fermi wavevector mismatch between the normal metal and the superconductor.
We argue that the topological charge density wave phase in the quasi-2D Kagome superconductor AV$_3$Sb$_5$ is a chiral flux phase. Considering the symmetry of the Kagome lattice, we show that the chiral flux phase has the lowest energy among those states which exhibit $2\times2$ charge orders observed experimentally. This state breaks the time-reversal symmetry and displays anomalous Hall effect. The explicit pattern of the density of this state in real space is calculated. These results are supported by recent experiments and suggest that these materials are a new platform to investigate the …
Chiral superconductors support chiral edge modes and potentially spontaneous edge currents at their boundaries. Motivated by the putative multiband chiral p-wave superconductor Sr$_2$RuO$_4$, we study the influence of the interference between different bands at the edges, which may appear in the presence of moderate edge disorder or in edge tunneling measurements. We show that interband interference can strongly modify the measurable quantities at the edges when the order parameter exhibits phase difference between the bands. This is illustrated by investigating the edge dispersion and the edg…
We study the collective behavior of Majorana modes in the vortex state of chiral $p$-wave superconductors. Away from the isolated vortex limit, the zero-energy Majorana states communicate with each other on a vortex lattice, and form a coherent band structure with non-trivial topological character. We revealed that the topological nature of Majorana bands changes sensitively via quantum phase transitions in the two-dimensional (2D) systems, as sweeping magnetic field or Fermi energy. Through the dimensional reduction, we showed the existence of generic superconducting Weyl phase in a low magne…
On the basis of Eilenberger theory, we study the vortex state when a magnetic field is applied nearly parallel to the ab plane in a chiral p-wave superconductor with a large anisotropy ratio of ab and c, as in Sr2RuO4. We quantitatively estimate the field dependence of the pair potential, magnetization, and flux line lattice form factor, and study the transition from the chiral p_- state at low fields to the nonchiral p_y state at high fields. Even for exactly parallel fields to the ab plane, transverse fields exist in the chiral state. The chiral-nonchiral transition disappears when the magne…
We discuss the origin of charge density wave (CDW) and spin density wave (SDW) in p-wave, d-wave and f-wave superconductors. To describe the low-energy quasiparticle excitation of p-wave case, we introduce a two- (one for time and one for space) dimensional massless Dirac model. After the non-Abelian bosonization is performed, the charge and spin density waves emerge from the model. By using this scheme, we try to explain the characteristic aspect of phase diagrams of various compounds, oxides and organic superconductors. The purpose of this paper is to make an argument that the dimensionality…
A simulation study for the magnetization process is performed for the multi-domain state in a chiral p-wave superconductor, using the time-dependent Ginzburg-Landau theory. The external field penetrates inside as core-less vortices through the domain wall, forming the vortex sheet structure. We find that, with increasing magnetic fields, the domain walls move so that the unstable domains shrink to vanish. Therefore, the single domain structure is realized at higher fields.
The edges of time reversal symmetry breaking topological superconductors support chiral Majorana bound states as well as spontaneous charge currents. The Majorana modes are a robust, topological property, but the charge currents are non-topological--and therefore sensitive to microscopic details--even if we neglect Meissner screening. We give insight into the non-topological nature of edge currents in chiral p-wave superconductors using a variety of theoretical techniques, including lattice Bogoliubov-de Gennes equations, the quasiclassical approximation, and the gradient expansion, and descri…
In a quasi-two-dimensional $ p $-wave superconductor there find six Cooper pairing states which are degenerate within the weak-coupling approach. We show that this degeneracy can be lifted by feedback effect favoring the so-called chiral $ p $-wave state. This effect is based on the anomalous coupling between charge and current in a system with broken time reversal symmetry and parity.
We consider the stability of the superconducting phase for spin-triplet p-wave pairing in a quasi-two-dimensional system. We show that in the absence of spin-orbit coupling there is a chiral contribution to spin fluctuation feedback which is related to spin quantum Hall effect in a chiral superconducting phase. We show that this mechanism supports the stability of a chiral p-wave state.
Observability of half-quantum vortices and skyrmions in p-wave superconductors is an outstanding open question. Under the most common conditions, fractional flux vortices are not thermodynamically stable in bulk samples. Here we show that in chiral p-wave superconductors, there is a regime where, in contrast lattices of integer flux vortices are not thermodynamically stable. Instead skyrmions made of spatially separated half-quantum vortices are the topological defects produced by an applied external field.
Chiral superconductors are two-fold degenerate and domains of opposite chirality can form, separated by domain walls. There are indications of such domain formation in the quasi two-dimensional putative chiral $p$-wave superconductor Sr$_2$RuO$_4$, yet no experiment has explicitly resolved individual domains in this material. In this work, $c$-axis domain walls lying parallel to the layers in chiral $p$-wave superconductors are explored from a theoretical point of view. First, using both a phenomenological Ginzburg-Landau and a quasiclassical Bogoliubov-deGennes approach, a consistent qualitat…
In order to identify the pairing symmetry with chirality, we study site-selective NMR in chiral p-wave superconductors. We calculate local nuclear relaxation rate 1/T_1 in the vortex lattice state by Eilenberger theory, including the applied magnetic field dependence. We find that 1/T_1 in the NMR resonance line shape is different between two chiral states p_{pm}(=p_x{pm}ip_y), depending on whether the chirality is parallel or anti-parallel to the vorticity. Anomalous suppression of 1/T_1 occurs around the vortex core in the chiral p_- wave due to the negative coherence term coming from the od…
