Chiral D-wave Research Articles

Majorana vortex modes in spin-singlet chiral superconductors with noncollinear spin ordering: Local density of states study

In the present study topologically nontrivial edge and vortex bound states are described in the coexistence phase of chiral spin-singlet superconductivity and noncollinear spin ordering on a triangular lattice in the presence of few (up to four) vortices. We consider the topological phase transition induced by the magnetic order between the phase hosting Majorana modes and the initial phase of the chiral d-wave superconductivity supporting non-Majorana modes which is also topologically nontrivial. The change of the excitation spectrum at the critical point is obtained in both cases of open and periodic boundary conditions in the presence of vortices. It is proved that zero energy Majorana modes localized at vortex cores are caused by noncollinear long-range magnetic ordering. Even though nearby excitation energies of subgap states including the edge-localized and vortex-localized states are very close to each other, the energy difference between different vortex bound states is an order of magnitude higher. This difference determines the energy gap for Majorana vortex modes separating them from other vortex bound states. It is found that even in the presence of noncollinear spin ordering its value can be estimated from the excitation energy of vortex bound states in the pure chiral d-wave state for the nonmagnetic case. By studying local density of states near the vortex cores the possibility to experimentally detect the described Majorana vortex modes by scanning tunneling microscopy is discussed. It is demonstrated that Majorana vortex modes and Majorana antivortex modes induced by noncollinear magnetism have different features in energy and spatially resolved density of states due to the chiral symmetry on the superconducting order parameter.

Evidence for chiral superconductivity on a silicon surface

Tin adatoms on a Si(111) substrate with a one-third monolayer coverage form a two-dimensional triangular lattice with one unpaired electron per site. These electrons order into an antiferromagnetic Mott-insulating state, but doping the Sn layer with holes creates a two-dimensional conductor that becomes superconducting at low temperatures. Although the pairing symmetry of the superconducting state is currently unknown, the combination of repulsive interactions and frustration inherent in the triangular adatom lattice opens up the possibility of a chiral order parameter. Here we study the superconducting state of Sn/Si(111) using scanning tunnelling microscopy, scanning tunnelling spectroscopy and quasiparticle interference imaging. We find evidence for a doping-dependent superconducting critical temperature with a fully gapped order parameter, the presence of time-reversal symmetry breaking and a strong enhancement in zero-bias conductance near the edges of the superconducting domains. Although each individual piece of evidence could have a more mundane interpretation, our combined results suggest the possibility that Sn/Si(111) is an unconventional chiral d-wave superconductor.

Local Kekulé distortion turns twisted bilayer graphene into topological Mott insulators and superconductors

Magic-angle twisted bilayer graphene displays at different fillings of the four flat bands lying around the charge neutrality point a wealth of notable phases that include magnetic Chern insulators, whose magnetization is mostly of an orbital nature and contiguous superconducting domes. Such a rich phase diagram is explained through the positive interplay of Coulomb repulsion and the electron coupling to a twofold optical mode that corresponds to Kekul\'e distortions localized into the small AA stacked regions of the moir\'e supercells. A static distortion stabilizes, at any integer filling of the flat bands, valence-bond insulators that carry finite Chern number away from charge neutrality. Similarly, a dynamic distortion that resonates between the two lattice vibrations leads to resonating-valence-bond topological insulators with built-in chiral d-wave pairs that have finite Chern number equal to the angular momentum, and thus are prone to turn superconducting upon doping away from integer filling.

Antiferromagnetism and chiral-[formula omitted] wave superconductivity in a honeycomb lattice close to Mott state

The antiferromagnetism (AFM) and chiral-d wave superconductivity (SC) in a honeycomb lattice close to an antiferromagnetic (AF) Mott state at half band filling are studied with a t−J model and slave boson mean field theory. The order parameters and single particle dispersion relations at different band filling fractions are investigated. It is found that the AFM enhances the SC, and leads to nodal single particle dispersion relations at two band filling fractions. These unexpected nodal dispersion relations out of nodeless chiral-d wave superconducting order are discussed. A comparison between AF chiral-d wave states and AF extended-s wave states is also given to highlight the pertinent features in chiral-d wave superconducting states. This study may be related to the honeycomb lattice materials such as the In3Cu2VO9 compound.

Local observation of linear- T superfluid density and anomalous vortex dynamics in URu2Si2

The heavy fermion superconductor URu$_2$Si$_2$ is a candidate for chiral, time-reversal symmetry-breaking superconductivity with a nodal gap structure. Here, we microscopically visualized superconductivity and spatially inhomogeneous ferromagnetism in URu$_2$Si$_2$. We observed linear-$T$ superfluid density, consistent with d-wave pairing symmetries including chiral d-wave, but did not observe the spontaneous magnetization expected for chiral d-wave. Local vortex pinning potentials had either four- or two-fold rotational symmetries with various orientations at different locations. Taken together, these data support a nodal gap structure in URu$_2$Si$_2$ and suggest that chirality either is not present or does not lead to detectable spontaneous magnetization.

Chiral singlet superconductivity in the weakly correlated metal LaPt3P

Chiral superconductors are novel topological materials with finite angular momentum Cooper pairs circulating around a unique chiral axis, thereby spontaneously breaking time-reversal symmetry. They are rather scarce and usually feature triplet pairing: a canonical example is the chiral p-wave state realized in the A-phase of superfluid He3. Chiral triplet superconductors are, however, topologically fragile with the corresponding gapless boundary modes only weakly protected against symmetry-preserving perturbations in contrast to their singlet counterparts. Using muon spin relaxation measurements, here we report that the weakly correlated pnictide compound LaPt3P has the two key features of a chiral superconductor: spontaneous magnetic fields inside the superconducting state indicating broken time-reversal symmetry and low temperature linear behaviour in the superfluid density indicating line nodes in the order parameter. Using symmetry analysis, first principles band structure calculation and mean-field theory, we unambiguously establish that the superconducting ground state of LaPt3P is a chiral d-wave singlet.

Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of strained graphene

We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for highly unconventional superconducting states with the coexistence of chiral d-wave superconductivity, charge density waves and pair density waves in the strained graphene. Our results show that the strain-induced flat bands can promote the superconducting transition temperature approximately 50% compared to that of the original doped graphene, which suggests that the flat-band superconductivity is a potential route to get superconductivity with higher critical temperatures. In particular, we obtain the superfluid weight for the pure superconducting pair-density-wave states from which the deduced superconducting transition temperature is shown to be much lower than the gap-opening temperature of the pair density wave, which is helpful to understand the phenomenon of the pseudogap state in high-Tc cuprate superconductors. Finally, we show that the BKT transition temperature versus doping for strained graphene exhibits a dome-like shape and it depends linearly on the spin-spin interaction strength.

Anisotropic Kondo pseudogap in URu2Si2

A polarized electronic Raman scattering study reveals the emergence of symmetry dependence in the electronic Raman response of single crystalline URu$_{2}$Si$_{2}$ below the Kondo crossover scale $T_K\sim100K$. In particular, the development of a coherent Kondo pseudo-gap predominantly in the E$_g$ channel highlights strong anisotropy in the Kondo physics in URu$_{2}$Si$_{2}$ that has previously been neglected in theoretical models of this system. A calculation of the Raman vertices demonstrates that the strongest Raman vertex does indeed develop within the E$_g$ channel for interband transitions and reaches a maximum along the diagonals of the Brillouin zone, implying a d-wave-like geometry for the Kondo pseudo-gap. Below the hidden order phase transition at $T_{HO}= 17.5K$, the magnitude of the pseudo-gap is found to be enhanced. Moreover, the anisotropy of the pseudo-gap is similar in form to that proposed for the chiral d-wave (E$_g$) superconducting state that appears below $T_c=1.5K$.

Spontaneous thermal Hall conductance in superconductors with broken time-reversal symmetry

This paper paper investigates the spontaneous thermal Hall conductivity in chiral d-wave superconductors. The authors show that the contribution of the impurity effect can be orders of magnitude larger than a possible topological contribution.

Antiferromagnetism and chiral d -wave superconductivity from an effective t−J−D model for twisted bilayer graphene

Starting from the strong-coupling limit of an extended Hubbard model, we develop a spin-fermion theory to study the insulating phase and pairing symmetry of the superconducting phase in twisted bilayer graphene. Assuming that the insulating phase is an anti-ferromagnetic insulator, we show that fluctuations of the anti-ferromagnetic order in the conducting phase can mediate superconducting pairing. Using a self-consistent mean-field analysis, we find that the pairing wave function has a chiral d-wave symmetry. Consistent with this observation, we show explicitly the existence of chiral Majorana edge modes by diagonalizing our proposed Hamiltonian on a finite-sized system. These results establish twisted bilayer graphene as a promising platform to realize topological superconductivity.

Unconventional chiral d-wave superconducting state in strained graphene**Project supported by the National Natural Science Foundation of China (Grant Nos. 11804213 and 11605109).

We consider a highly unconventional superconducting state with chiral d-wave symmetry in doped graphene under strain with the Gutzwiller–RVB method in the momentum space. It is shown that flat bands emerge in the normal state for reasonable strain. As a result, the superconducting critical temperature is found to be linearly proportional to the strength of the electron–electron interaction. Furthermore, the chiral d-wave superconducting state is shown with coexistence of the charge density wave and the pair density wave. There are different coexisting states with those orders under different doping levels.

Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands.

Flexible long period moiré superlattices form in two-dimensional van der Waals crystals containing layers that differ slightly in lattice constant or orientation. In this Letter we show theoretically that isolated flat moiré bands described by generalized triangular lattice Hubbard models are present in twisted transition metal dichalcogenide heterobilayers. The hopping and interaction strength parameters of the Hubbard model can be tuned by varying the twist angle and the three-dimensional dielectric environment. When the flat moiré bands are partially filled, candidate many-body ground states at some special filling factors include spin-liquid states, quantum anomalous Hall insulators, and chiral d-wave superconductors.

Probing unconventional superconductivity in proximitized graphene by impurity scattering

We demonstrate how potential impurities are a very powerful tool for determining the pairing symmetry in graphene proximity-coupled to a spin-singlet superconductor. All d-wave states are characterized by subgap resonances, with spatial patterns clearly distinguishing between nodal and chiral d-wave symmetry, while s-wave states have no subgap resonances. We also find strong supergap impurity resonances associated with the normal state Dirac point. Sub- and supergap resonances only interact at very low doping levels, then causing suppression of the supergap resonances.

Unconventional pairing symmetry of interacting Dirac fermions on a π -flux lattice

The pairing symmetry of interacting Dirac fermions on the $\pi$-flux lattice is studied with the determinant quantum Monte Carlo and numerical linked cluster expansion methods. The extended $s^*$- (i.e. extended $s$-) and d-wave pairing symmetries, which are distinct in the conventional square lattice, are degenerate under the Landau gauge. We demonstrate that the dominant pairing channel at strong interactions is an exotic $ds^*$-wave phase consisting of alternating stripes of $s^*$- and d-wave phases. A complementary mean-field analysis shows that while the $s^*$- and d-wave symmetries individually have nodes in the energy spectrum, the $ds^*$ channel is fully gapped. The results represent a new realization of pairing in Dirac systems, connected to the problem of chiral d-wave pairing on the honeycomb lattice, which might be more readily accessed by cold-atom experiments.

Chiral d -wave superconductivity in a triangular surface lattice mediated by long-range interaction

Correlated ad-atom systems on the Si(111) surface have recently attracted an increased attention as strongly correlated systems with a rich phase diagram. We study these materials by a single band model on the triangular lattice including 1/r long-range interaction. Employing the recently proposed TRILEX method we find an unconventional superconducting phase of chiral d-wave symmetry in hole-doped systems. The superconductivity is driven simultaneously by both charge and spin fluctuations and is strongly enhanced by the long-range tail of the interaction. We provide an analysis of the relevant collective bosonic modes and explain how in triangular symmetry both charge and spin channels contribute to the Cooper-pairing.

The nontrivial ground state topology in the coexistence phase of chiral d-wave superconductivity and 120-degree magnetic order on a triangular lattice

The Z2 topological invariant is defined in the chiral d-wave superconductor having a triangular lattice in the presence of the 120-degree magnetic ordering. Analyzing the Z2 invariant, we determine the conditions of implementing topological phases in the model with regard to superconducting pairings between the nearest and next nearest neighbors. It is often supposed in such a system that the pairing parameter between the nearest neighbors should be equal to zero due to the intersite Coulomb interaction. We show that taking into account even weak pairings in the first coordination sphere leads to the disappearance of the gapless excitations of the bulk spectrum in the wide region of the parameter space. Thus, topological invariants can be defined in this region. In solving the problem of open edges it is shown that the zero energy modes are realized basically in the topologically nontrivial phases. Such zero modes are topologically protected Majorana modes. A connection between the Z2 invariant and the integer topological invariant of the ground state of the 2D lattice is established in the presence of the electron–hole symmetry and noncollinear magnetic ordering.

Surface magnetism in a chiral d -wave superconductor with hexagonal symmetry

Surface properties are examined in a chiral d-wave superconductor with hexagonal symmetry, whose one-body Hamiltonian possesses the intrinsic spin-orbit coupling identical to the one characterizing the topological nature of the Kane-Mele honeycomb insulator. In the normal state spin-orbit coupling gives rise to spontaneous surface spin currents, whereas in the superconducting state there exist besides the spin currents also charge surface currents, due to the chiral pairing symmetry. Interestingly, the combination of these two currents results in a surface spin polarization, whose spatial dependence is markedly different on the zigzag and armchair surfaces. We discuss various potential candidate materials, such as SrPtAs, which may exhibit these surface properties.

Andreev reflection in two-dimensional relativistic materials with realistic tunneling transparency in normal-metal/superconductor junctions

The Andreev conductance across 2d normal metal (N)/superconductor (SC) junctions with relativistic Dirac spectrum is investigated theoretically in the Blonder-Tinkham-Klapwijk formalism. It is shown that for relativistic materials, due to the Klein tunneling instead of impurity potentials, the local strain in the junction is the key factor that determines the transparency of the junction. The local strain is shown to generate an effective Dirac $\delta$-gauge field. A remarkable suppression of the conductance are observed as the strength of the gauge field increases. The behaviors of the conductance are in well agreement with the results obtained in the case of 1d N/SC junction. We also study the Andreev reflection in a topological material near the chiral-to-helical phase transition in the presence of a local strain. The N side of the N/SC junction is modeled by the doped Kane-Mele (KM) model. The SC region is a doped correlated KM t-J (KMtJ) model, which has been shown to feature d+id'-wave spin-singlet pairing. With increasing intrinsic spin-orbit (SO) coupling, the doped KMtJ system undergoes a topological phase transition from the chiral d-wave superconductivity to the spin-Chern superconducting phase with helical Majorana fermions at edges. We explore the Andreev conductance at the two inequivalent Dirac points, respectively and predict the distinctive behaviors for the Andreev conductance across the topological phase transition. Relevance of our results for the adatom-doped graphene is discussed.

Topological orbital superfluid with chiral d-wave order in a rotating optical lattice

Topological superfluid is an exotic state of quantum matter that possesses a nodeless superfluid gap in the bulk and Andreev edge modes at the boundary of a finite system. Here, we study a multi-orbital superfluid driven by an attractive s-wave interaction in a rotating optical lattice. Interestingly, we find that the rotation induces the inter-orbital hybridization and drives the system into topological orbital superfluid in accordance with intrinsically chiral d-wave pairing characteristics. Thanks to the conservation of spin, the topological orbital superfluid supports four rather than two chiral Andreev edge modes at the boundary of the lattice. Moreover, we find that the intrinsic harmonic confining potential forms a circular spatial barrier which accumulates atoms and supports a mass current under the injection of small angular momentum as an external driving force. This feature provides an experimentally detectable phenomenon to verify the topological orbital superfluid with chiral d-wave order in a rotating optical lattice.

Identifying the chiral d-wave superconductivity by Josephson \u03c60-states

We propose the Josephson junctions linked by a normal metal between a d + id superconductor and another d + id superconductor, a d-wave superconductor, or a s-wave superconductor for identifying the chiral d + id superconductivity. The time-reversal breaking in the chiral d-wave superconducting state is shown to result in a Josephson φ0-junction state where the current-phase relation is shifted by a phase φ0 from the sinusoidal relation, other than 0 and π. The ground-state phase difference φ0 and the critical current can be used to definitely confirm and read the information about the d + id superconductivity. A smooth evolution from conventional 0-π transitions to tunable φ0-states can be observed by changing the relative magnitude of two types of d-wave components in the d + id pairing. On the other hand, the Josephson junction involving the d + id superconductor is also the simplest model to realize a φ0- junction, which is useful in superconducting electronics and superconducting quantum computation.