Session Z01: Topological Superconductivity: Junctions and Spectroscopy

Inhomogeneity-Induced Time-Reversal Symmetry Breaking in Cuprate Twist-Junctions

The lowest order Josephson coupling, J1(θ)cos(ϕ), between two d-wave superconductors with phase-difference ϕ across the junction vanishes when their relative orientation is rotated by θ=π/4. However, in the presence of inhomogeneity, J1(r) is non-zero locally, with a sign that fluctuates in space. We show that such a random J1 generates a global second-harmonic Josephson coupling, J2cos(2ϕ), with a sign that favors ϕ=±π/2, i.e., spontaneous breaking of time reversal symmetry. The magnitude of J2 is substantially enhanced if the spatial correlations of J1(r) extend over large distances, such as would be expected in the presence of large amplitude twist-angle angle disorder or significant local electronic nematicity. We argue that this effect likely accounts for the recent observations in twisted Josephson junctions between high temperature superconductors.   

Topological Phase Transitions in 2.5 D Twisted Nodal Superconductors

Under the application of an interlayer supercurrent, twisted bilayers of nodal superconductors have been predicted to host topological states. Experiments on cuprate materials, however, are so far limited to finite thickness flakes containing many layers, raising the question how topological effects may be altered. We demonstrate that in finite-thickness flakes experimental signatures of topological superconductivity remain robust and new topological transitions arise, that have no analogue in bilayers. 

We study an N layered flake of d_x²-y² superconductor with only one (top) layer twisted with respect to the others, and a c-axis supercurrent through the system. For a weak current, the low-energy quasiparticle dispersion contains 4N gapped Dirac nodes with a total Chern number of |C| = 2N. The average gap size decreases as O(1/N), but at large twist angles the gap of the top layer remains robust to increasing N, making this regime promising for scanning tunneling microscopy experiments. Increasing the current leads to a sequence of topological transitions, eventually increasing the Chern number to be of the order O(N²). We discuss the signatures of these transitions in thermal Hall effect, demonstrating that it is robust to increasing layer thickness at intermediate temperatures.   

Majorana bound states in a d-wave superconductor planar Josephson junction

We study phase-controlled planar Josephson junction comprising a two-dimensional electron gas with strong spin-orbit coupling and d-wave superconductors, which have an advantage of high critical temperature. We show that a region between the two superconductors can be tuned into topological state by the in-plane Zeeman field, and can host Majorana bound states. We further investigate the behavior of the topological gap and its dependence on the type of d-wave pairing, i.e., d, d + is, or d + id′, and note the difficulties that can arise due to the presence of gapless excitations in pure d-wave superconductors. In case of d + is pairing, we have demonstrated realizations of MBS for a wide range of parameters. In the case of d + id′ pairing there are no gapless states in the bulk of 2DEG; however, the even Chern number associated with the bulk leads to appearance of gapless chiral edge modes, which can hybridize with MBS. To realize MBS with d + id′ pairing, we have proposed a modified JJ in which the chiral edge modes are gapped and do not hybridize with MBS. Our proposal can be realized in cuprate superconductors, e.g., in a twisted bilayer, combined with the layered semiconductor Bi2O2Se.   

Diode effect in a d-wave superconductor planar Josephson junction

We consider phase-controlled planar Josephson junction comprising a two-dimensional electron gas with strong spin-orbit coupling and d-wave superconductors.

We investigate the superconducting diode effect that originates from a mirror and time-reversal symmetry breaking in the above planar Josephson junction. Josephson's current is calculated analytically and numerically for different pairing orientations using Matsubara Green's function approach. We also extend our analysis to d+id', d+is pairings realizable in twisted cuprate bilayers. The nonreciprocity is quantified by the quality factor $Q = frac{I_{c}^R-I_{c}^L}{I_c^R+I_c^L}$, where $I_c^R(I_c^L)$ refers to the maximum forward (backward) Josephson current, for different system parameters, including the Zeeman-field, spin-orbit coupling, pairing orientation, and gate bias. Furthermore, we try to differentiate between the super-current that arises due to the presence of edge states and the bulk states. We optimize the quality factor for various system parameters including the length of the junction and d-wave pairing angle. Finally, we discuss the relevance of Majorana-bound states and how they influence the observed nonreciprocity.     

Superconducting diode effect in 1T’-WS2/2H-WS2 heterophase bilayer

The nonreciprocity of the superconducting critical current is known as the superconducting diode effect, which can be realized in noncentrosymmetric superconductors with broken time reversal symmetry. Using density functional theory, we propose a new system, the 1T’-WS₂/2H-WS₂ heterophase bilayer, that exhibits the superconducting diode effect. We find that the electron doping from the 2H-WS₂ layer to the 1T’-WS₂ layer creates an interface dipole and induces Rashba band splitting. This results in the formation of finite momentum Cooper pairs under an in-plane magnetic field that lead to nonreciprocity of critical current. Using ab initio Migdal–Eliashberg theory, we confirm the superconductivity in this heterophase bilayer. We also found that the critical temperature of the heterophase bilayer depends on the strain and the amount of charge transfer. Our work demonstrates that by exploring different combinations of TMDC materials and phases, one can achieve novel topological superconducting properties in van der Waals heterostructures of 2D materials.   

Unveiling Quasiparticle Berry Curvature Effects in the Spectroscopic Properties of a Topological Superconductor

The focus of experimental efforts on topological superconductivity (Tsc) has predominantly centered around the detection of Majorana boundary modes. On the contrary, the experimental features arising from the band topology in the bulk remain relatively unexplored. In this talk, I will discuss how spectroscopic properties away from the boundaries can be influenced by the Berry curvatures (BC) of Bogoliubov quasiparticle in a chiral p-wave Tsc achieved by spin-orbit coupling. Beyond the more well-known momentum-space BC effect, I will show that a new phase-space BC term emerges under an applied magnetic field, exclusively when the spin-orbit coupling and superconductivity coexist. Importantly, this novel BC term leads to qualitative modifications in energy- and momentum-resolved tunneling spectroscopy. Such a BC effect can be sizably amplified by the strength of spin-orbit coupling and the applied field, offering a new potentially detectable experimental feature in a chiral p-wave Tsc.   

Nonlinear Optical Responses in Multi-Orbital Topological Superconductors

We study the non-linear optical responses of a multi-band noncentrosymmetric superconductor.  The absence of inversion symmetry in noncentrosymmetric superconductors allows for the coexistence of opposite parity pairing channels, thus leading to a mixed-parity order parameter with mixed-parity pairing terms. These factors can give rise to exotic properties, including topological superconductivity. The nature of the superconducting phase is determined by the intricate interplay of Rashba spin-orbit coupling and the inter- and intra-orbital pairing amplitudes. These aspects of the system can be utilized to drive a phase transition from trivial to topological superconducting phases. We explore the suitability of the second-order conductivity as a probe for studying the topological nature of the superconducting phase of the system. We find that the second-order response, in contrast with the linear response, contains clear signatures that can be used to differentiate between trivial and topological superconducting phases. In particular, the low-frequency second-order DC response, also known as the photogalvanic response, exhibits distinctive features in different phases, indicating that it can serve as a potential probe to determine the topological nature of the superconducting state.   

Unveiling Majorana oscillations with temperature effects

Distinguishing trivial and topological phases in Majorana wires is challenging due to, among

others, the ubiquitous presence of disorder-driven trivial states. In this work [1], we propose a

method of observing Majorana oscillations that trivial states are unable to mimic. To this end, we

calculate the transport and spectral properties of Majorana wires by using the scattering matrix

in the Bogoliubov-de Gennes formalism. We find that for experimentally accessible parameter

regimes and electron temperatures of the order of 50mK or higher, we are able to observe Majorana

oscillations in variations of local conductance when we change the coupling to one of the leads,

i.e. δGLL = GLL(ΓR = ΓL) − GLL(ΓR << ΓL), where ΓL,R represent the couplings to the leads.

Importantly, these oscillations are only observed in the topological phase, as opposed to oscillations

in GLL, which can be mimicked, for instance, by quasi-Majoranas. We investigate the origin of such

correlation between left and right sides in terms of the wavefunction profiles of the Majorana modes

and the hybridization between them. Our proposal provides a method of distinguishing topological

and trivial subgap states and may provide guidance for future experiments in the field.

[1] R.A. Dourado, P.H. Penteado and J.C. Egues, “Nonlocality of local Andreev conductances as a probe for

topological Majorana wires”, arXiv:2303.01867 (2023).   

Pairing Symmetry and Projective Symmetry Groups for Fermions

Determining the pairing symmetry in a superconductor is a question of both conceptual and practical importance. In this work, we elucidate the connection between pairing symmetry of a superconductor and the fermion projective symmetry group (PSG). The normal state before the superconducting pairing formation has the symmetry X´U(1), where X denotes the crystalline symmetry. The formation of superconducting pairing breaks the U(1) charge conservation down to $Z_2^f$ fermion parity symmetry. The correct symmetry group for fermions in the superconducting phase is therefore a central extension of X by $Z_2^f$, mathematically described by the fermion PSG. It turns out that the representation of the pairing order parameter is intimately related to the PSGs of the fermions. We studied in detail the correspondence between fermion PSGs and the superconductor pairing symmetry for all point group symmetries. The fermion PSG constrains many other properties of the system, such as selection rules in Raman spectroscopy, classification of topological superconductors, and the presence/absence of Majorana zero modes at vortex cores. Our work provides a general framework to determine pairing symmetry through the study of fermion PSGs.   

Pairing Symmetry and Classification of Majorana Zero Modes in a Superconducting Vortex

We present a complete topological classification of Majorana Zero Modes (MZMs) bound to the core of a superconducting vortex in 2D in the presence of additional crystalline symmetries. For each relevant magnetic point group G, we first find all the fermion symmetry groups G_f that are characterized by inequivalent central extensions of G with respect to fermion parity Z^f₂. For each G_f we employ Kitaev’s K-theory methodology, together with the Teo-Kane approach for topological defects, to classify the MZM multiplets using the bulk topological index of an effective one-dimensional chain with on-site symmetries. We analyze systems with both weak and strong spin orbit coupling. We also study the effect of interactions on the non-interacting Bogoliubov-deGennes classification. We observe that crystalline symmetries can protect MZM multiplets at the vortex core which can otherwise be gapped out. We discuss the close correspondence between pairing symmetry, G_f and MZM classification which, in certain symmetry classes, can make MZMs a potential probe of pairing symmetry.    

Optical and Raman selection rules for the pairing symmetry in chiral superconductors

Unconventional superconductivity has non-s-wave pairing order parameters, whose pairing symmetries are often difficult to determine experimentally. We propose a new method to probe the pairing symmetry with optical spectroscopy. Different pairing symmetries correspond to distinct electron symmetry fractionalization, leading to different quantum numbers of optical excitations. This gives rise to selection rules for the particle-hole continuum in optical/Terahertz absorption spectroscopy, and Raman scattering spectroscopy. We apply this approach to chiral superconductors with strong spin-orbit couplings, and to singlet superconductors with weak spin-orbit couplings, where the parity of the pairing order parameters can be determined unambiguously from spectroscopy measurements. We demonstrate the signature of these selection rules in optical spectra using specific examples.