Odd-parity Superconductivity Research Articles

Exposing the odd-parity superconductivity in CeRh2As2 with hydrostatic pressure

Odd-parity superconductivity is a fundamentally interesting but rare state of matter with a potential for applications in topological quantum computing. Crystals with staggered locally noncentrosymmetric structures have been proposed as platforms where a magnetic field can induce a transition between even- and odd-parity superconducting (SC) states. The strongly correlated superconductor CeRh2As2, with the critical temperature Tc≈0.4K, is likely the first example material showing such a phase transition, which occurs at the magnetic field μ0H*=4T applied along the crystallographic c axis. CeRh2As2 also undergoes a phase transition of an unknown origin at T0=0.5K. By subjecting CeRh2As2 to hydrostatic pressure and mapping the resultant changes to the SC phase diagrams we investigated how the lattice compression and changes to the electronic correlations affect the stability and relative balance of the two SC states. The abnormally high in-plane upper critical field becomes even higher close to a quantum critical point of the T0 order. Remarkably, the SC phase-switching field H* is drastically reduced under pressure, dropping to 0.3 T at 2.7 GPa. This result signals an apparent strengthening of the local noncentrosymmetricity and forecasts a possible stabilization of the putative odd-parity state down to zero field, hitherto not considered by theoretical models. Published by the American Physical Society 2024

Optical and Raman selection rules for odd-parity clean superconductors

Optical and Raman selection rules for odd-parity clean superconductors

Fermi Surface Spin Texture and Topological Superconductivity in Spin-Orbit Free Noncollinear Antiferromagnets.

We explore the relationship among the magnetic ordering in real space, the resulting spin texture on the Fermi surface, and the related superconducting gap structure in noncollinear antiferromagnetic metals without spin-orbit coupling. Via a perturbative approach, we show that noncollinear magnetic ordering in a metal can generate momentum-dependent spin texture on its Fermi surface, even in the absence of spin-orbit coupling, if the metal has more than three sublattices in its magnetic unit cell. Thus, our theory naturally extends the idea of altermagnetism to noncollinear spin structures. When superconductivity is developed in a magnetic metal, as the gap-opening condition is strongly constrained by the spin texture, the nodal structure of the superconducting state is also enforced by the magnetism-induced spin texture. Taking the noncollinear antiferromagnet on the kagome lattice as a representative example, we demonstrate how the Fermi surface spin texture induced by noncollinear antiferromagnetism naturally leads to odd-parity spin-triplet superconductivity with nontrivial topological properties.

Observation of Odd-Parity Superconductivity with the Geshkenbein-Larkin-Barone Composite Rings.

Phase-sensitive measurements on a composite ring made of a superconductor of interest connected by a known singlet s-wave superconductor can unambiguously determine its pairing symmetry. In composite rings with epitaxial β-Bi_{2}Pd and s-wave Nb, we have observed half-integer-quantum flux when Nb is connected to the opposite crystalline ends of β-Bi_{2}Pd and integer-quantum flux when Nb is connected to the same crystalline ends of β-Bi_{2}Pd. With ascending temperature, the half-integer-flux quantization transits to integer-flux quantization, before the eventual loss of phase coherence. These findings point to odd-parity pairing symmetry in superconducting β-Bi_{2}Pd.

Hybrid-order topological odd-parity superconductors via Floquet engineering

Having the potential for performing quantum computation, topological superconductors have been generalized to the second-order case. The hybridization of different orders of topological superconductors is attractive because it facilitates the simultaneous utilization of their respective advantages. However, previous studies found that they cannot coexist in one system due to the constraint of symmetry. We propose a Floquet engineering scheme to generate two-dimensional (2D) hybrid-order topological superconductors in an odd-parity superconductor system. Exotic hybrid-order phases exhibiting the coexisting gapless chiral edge states and gapped Majorana corner states not only in two different quasienergy gaps but also in one single quasienergy gap are created by periodic driving. The generalization of this scheme to the 3D system allows us to discover a second-order Dirac superconductor featuring coexisting surface and hinge Majorana Fermi arcs. Our results break a fundamental limitation on topological phases and open a feasible avenue to realize topological superconductor phases without static analogs.

Microscopic Imaging Homogeneous and Single Phase Superfluid Density in UTe_{2}.

Odd-parity superconductor UTe_{2} shows spontaneous time-reversal symmetry breaking and multiple superconducting phases, which imply chiral superconductivity, but only in a subset of samples. Here we microscopically observe a homogeneous superfluid density n_{s} on the surface of UTe_{2} and an enhanced superconducting transition temperature near the edges. We also detect vortex-antivortex pairs even at zero magnetic field, indicating the existence of a hidden internal field. The temperature dependence of n_{s}, determined independent of sample geometry, does not support point nodes along the b axis for a quasi-2D Fermi surface and provides no evidence for multiple phase transitions in UTe_{2}.

Customizing topological phases in the twisted bilayer superconductors with even-parity pairings

We investigate the topological properties of twisted bilayer superconductors with different even-parity pairings in each layer. In the presence of spin–orbit coupling, the Hamiltonian is mapped into an effective odd-parity superconductor. Based on this, we deduce the topological properties by examining the relative configuration between Fermi surface and Dirac pairing node. We show that mixed Rashba and Dresselhaus spin–orbit coupling and anisotropic hopping terms, which break the C 4 symmetry of the Fermi surface, can induce first-order topological superconductors with non-zero bulk Chern number. This provides a versatile way to control the topological phases of bilayer superconductors by adjusting the twisted angle and chemical potential. We demonstrate our results using a typical twisted angle of 53.13°, at which the translation symmetry is restored and the Chern number and edge state are calculated using the Moiré momentum.

Nuclear spin relaxation rate of nonunitary Dirac and Weyl superconductors

Nonunitary superconductivity has attracted renewed interest as a novel gapless phase of matter. In this study, we investigate the superconducting gap structure of nonunitary odd-parity chiral pairing states in a superconductor involving strong spin-orbit interactions. By applying a group theoretical classification of chiral states in terms of discrete rotation symmetry, we categorized all possible point-nodal gap structures in nonunitary chiral states into four types in terms of the topological number of nodes and node positions relative to the rotation axis. In addition to conventional Dirac and Weyl point nodes, we identify a type of Dirac point nodes unique to nonunitary chiral superconducting states. Such a Dirac point node is a nodal point of two pseudospin excitations, but the dispersions around it depend on the pseudospin. The node type can be identified experimentally based on the temperature dependence of the nuclear magnetic resonance longitudinal relaxation rate. The implication of our results for a nonunitary odd-parity superconductor in ${\mathrm{UTe}}_{2}$ is also discussed.

Pair breaking in superconductors with strong spin-orbit coupling

We study the influence of symmetry-breaking perturbations on superconductivity in multiorbital materials, with a particular focus on an external magnetic field. We introduce the field-fitness function which characterizes the pair-breaking effects of the perturbation on a given superconducting state. For even-parity superconductors we find that this field-fitness function for an external magnetic field is one, implying that the paramagnetic response is controlled only by a generalized effective $g$ factor. For odd-parity superconductors, the interplay of the effective $g$ factor and the field-fitness function can lead to counterintuitive results. We demonstrate this for $p$-wave pairing in the effective $j=\frac{3}{2}$ electronic states of the Luttinger-Kohn model.

Degeneracy between even- and odd-parity superconductivity in the quasi-one-dimensional Hubbard model and implications for Sr2RuO4

Based on a weak coupling calculation, we show that an accidental degeneracy appears between even- and odd-parity superconductivity in the quasi-one-dimensional (1D) limit of the repulsive Hubbard model on the square lattice. We propose that this effect could be at play on the quasi-1D orbitals Ru ${d}_{zx}$ and ${d}_{zy}$ of ${\text{Sr}}_{2}{\text{RuO}}_{4}$, leading to a gap of the form ${\mathrm{\ensuremath{\Delta}}}_{\text{even}}+i{\mathrm{\ensuremath{\Delta}}}_{\text{odd}}$ which could help reconcile several experimental results.

Anomalous electromagnetic response in the spin-triplet superconductor UTe2

The recently discovered heavy-fermion superconductor ${\mathrm{UTe}}_{2}$, as a promising candidate for an odd-parity superconductivity, exhibits anomalous magnetic-field reinforced superconductivity with a huge upper critical field of 35 T. However, the thermodynamic properties of this unconventional superconductivity still remain unclear. Herein we present the results of dc magnetization and heat capacity measurements in ${\mathrm{UTe}}_{2}$ to probe the magnetic response at low temperatures. The obtained isothermal magnetization in the superconducting state of ${\mathrm{UTe}}_{2}$ clearly demonstrates the anomalous magnetic flux pinning effect, which is most likely related to the peculiar electromagnetic response of the spin-triplet pairing state in the hard-magnetization axis (orthorhombic $b$ axis). The ac susceptibility measurements support the anomalous vortex states in ${\mathrm{UTe}}_{2}$. We also found evidence for the Lifshitz transition just above ${H}_{\mathrm{c}2}$ and the anomalous flux pinning anomalies for the easy-magnetization axis ($a$ axis).

Multicomponent odd-parity superconductivity in UAu2 at high pressure

We report that high-quality single crystals of the hexagonal heavy fermion material uranium diauride (UAu2) become superconducting at pressures above 3.2 GPa, the pressure at which an unusual antiferromagnetic state is suppressed. The antiferromagnetic state hosts a marginal fermi liquid and the pressure evolution of the resistivity within this state is found to be very different from that approaching a standard quantum phase transition. The superconductivity that appears above this transition survives in high magnetic fields with a large critical field for all field directions. The critical field also has an unusual angle dependence suggesting that the superconductivity may have an order parameter with multiple components. An order parameter consistent with these observations is predicted to host half-quantum vortices (HQVs). Such vortices can be topologically entangled and have potential applications in quantum computing.

Time-reversal invariant topological gapped phases in bilayer Dirac materials

We show that a plethora of topological insulating phases can appear in bilayer Dirac materials, including first-order topological insulators, topological mirror insulators, and second-order topological insulators. By considering doping and short-range attractive interactions, we show at a mean-field level that intrinsic odd-parity superconductivity can arise in such systems. Depending on the number and positions of Fermi surfaces in the normal state, we find that the resulting odd-parity superconductivity can lead to diverse time-reversal invariant topological superconducting phases, including topological superconductors and topological mirror superconductors. Our findings suggest that bilayer Dirac materials could be an excellent platform for investigating topological phases.

Optical investigation of the heavy-fermion normal state in superconducting UTe2

The recently discovered superconductor, UTe$_2$, has attracted immense scientific interest due to the experimental observations that suggest odd-parity superconductivity. It is believed that the material becomes a heavy-fermion metal at low temperatures although details of the normal state are unclear. Using Fourier transform infrared spectroscopy (FTIR), the normal state electronic structure of UTe$_2$ was investigated at zero applied magnetic field. Combining the measured reflectivity with the dc resistivity, the complex optical conductivity was obtained over a large frequency range. The frequency dependence of the real part of the optical conductivity exhibits a MIR peak around 4000 cm$^{-1}$ and a narrow Drude peak that develops below 40 K. A combination of density functional and dynamic mean field theory (DFT + DMFT) gives spectra in close correspondence to the experiment. Via this comparison we attribute the prominent MIR peak to inter-band transitions involving a narrow U 5$f$ feature that develops near the Fermi level. In this regard, our data gives spectroscopic evidence for the existence of a low energy Kondo resonance at temperatures just above the onset of superconductivity and implicates heavy electrons in the formation of the superconducting state. We find that the coherent Kondo resonance is primarily associated with a collapse of scattering and less with a transfer of spectral weight.

Nonsymmorphic symmetry and field-driven odd-parity pairing in CeRh2As2

CeRh${}_{2}$As${}_{2}$ has attracted attention for a field-induced transition between two distinct superconducting states. The authors argue that the nonsymmorphic symmetry of the crystal lattice enables this. This symmetry enforces the spin polarization of the Bloch states at the Brillouin zone edges to be parallel to the edges. Such a spin polarization ensures a near-degeneracy between even- and odd-parity superconducting channels, provided the Fermi surface sits near these edges. First-principles calculations incorporating electronic correlations reveal a Fermi surface consistent with this picture.

Higher-order Weyl superconductors with anisotropic Weyl-point connectivity

Weyl superconductors feature Weyl points at zero energy in the three-dimensional (3D) Brillouin zone and arc states that connect the projections of these Weyl points on the surface. We report that higher-order Weyl superconductors can be realized in odd-parity topological superconductors with time-reversal symmetry being broken by periodic driving. Different from conventional Weyl points, the higher-order Weyl points in the bulk separate 2D first- and second-order topological phases, while on the surface, their projections are connected not only by conventional surface Majorana arcs, but also by hinge Majorana arcs. We show that the Weyl-point connectivity via Majorana arcs is largely enriched by the underlying higher-order topology and becomes anisotropic with respect to surface orientations. We identify the anisotropic Weyl-point connectivity as a characteristic feature of higher-order Weyl materials. As each 2D subsystem can be singled out by fixing the periodic driving, we propose how the Majorana zero modes in the 2D higher-order topological phases can be detected and manipulated in experiments.

Chiral Majorana hinge modes in superconducting Dirac materials

Chiral Majorana hinge modes are characteristic of a second-order topological superconductor in three dimensions. Here we systematically study pairing symmetry in the point group D_{2h}, and find that the leading pairing channels can be of s-, d-, and s+id-wave pairing in Dirac materials. Except for the odd-parity s-wave pairing superconductivity, the s+id-wave pairing superconductor is topologically nontrivial and possesses Majorana hinge and surface modes. The chiral Majorana hinge modes can be characterized by a winding number of the quadrupole moment, or quantized quadruple moment at the symmetrically invariant point. Our findings suggest the strong spin-orbital coupling, crystalline symmetries and electron-electron interaction in the Dirac materials may provide a microscopic mechanism to realize chiral Majorana hinge modes without utilizing the proximity effect or external fields.

Faithful derivation of symmetry indicators: A case study for topological superconductors with time-reversal and inversion symmetries

Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in $d-2$ or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of the bulk bands and can diagnose boundary features. Specifically, to obtain indicators manifesting clear bulk-boundary correspondence, we combine the topological crystal classification scheme in the real space and a twisted equivariant K group analysis in the momentum space. The key step is to disentangle the generally mixed strong and weak indicators through a systematic basis-matching procedure between our real-space and momentum-space approaches. We demonstrate our protocol using an example of two-dimensional time-reversal odd-parity superconductors, where the inversion symmetry is known to protect a higher-order phase with corner Majoranas. Symmetry indicators derived from our protocol can be readily applied to ab initio database and could fuel material predictions for strong and weak topological crystalline superconductors with various boundary features.

Odd-Parity Spin-Triplet Superconductivity in Centrosymmetric Antiferromagnetic Metals.

We propose a route to achieve odd-parity spin-triplet (OPST) superconductivity in metallic collinear antiferromagnets with inversion symmetry. Owing to the existence of hidden antiunitary symmetry, which we call the effective time-reversal symmetry (eTRS), the Fermi surfaces of ordinary antiferromagnetic metals are generally spin degenerate, and spin-singlet pairing is favored. However, by introducing a local inversion symmetry breaking perturbation that also breaks the eTRS, we can lift the degeneracy to obtain spin-polarized Fermi surfaces. In the weak-coupling limit, the spin-polarized Fermi surfaces constrain the electrons to form spin-triplet Cooper pairs with odd parity. Interestingly, all the odd-parity superconducting ground states we obtained host nontrivial band topologies manifested as chiral topological superconductors, second-order topological superconductors, and nodal superconductors. We propose that double perovskite oxides with collinear antiferromagnetic or ferrimagnetic ordering, such as SrLaVMoO_{6}, are promising candidate systems where our theoretical ideas can be applied to.

Surface chiral superconductivity in odd-parity nematic superconductors with magnetic impurities

We study odd-parity nematic superconductivity in doped topological insulators in presence of surface magnetic impurities. The peculiar surface subgap spectrum, characterized by a Majorana flatband, nodal cones and the surface states of the parent topological insulator, gives rise to an overall ferromagnetic RKKY interactions between the surface impurities. An additional coupling between the impurities and a preemptive chiral order parameter promote a surface time-reversal symmetry breaking solution at the surface of the system. We discuss the relevant scenarios and suggest to engineer surface chiral superconductivity by properly choosing magnetic adatoms with highly anisotropic exchange coupling.