Real-Band Topology in Superconductors and Superfluids

In systems with space-time inversion symmetries, a distinct type of topological structure known as real-band topology can emerge. Unlike conventional complex-band topology, real-band topology is defined for real Bloch wave functions, and it is characterized by topological invariants such as the Euler class and the Stiefel-Whitney classes. This framework has recently attracted growing interest in the study of twisted bilayer graphene, topological insulators, semimetals, and metamaterials.

Our research explores how real-band topology arises in superconductors and superfluids. We first demonstrated that, under appropriate space–time symmetries, the Bogoliubov-de Gennes Hamiltonian of a superconductor can also host real-band topology. We then derived general relations connecting these real-band topological invariants to other known topological invariants. Using these results, we showed that the B phase of superfluid helium-3 in a magnetic field is characterized by a nontrivial Euler class and therefore realizes a topological superfluid protected by real-band topology.

The B phase of superfluid helium-3 is already known to be a topological superfluid in the presence of time-reversal symmetry. However, once a magnetic field is applied, this conventional topological invariant is no longer defined. Despite this, several studies have reported robust topological phenomena even in the presence of a magnetic field, leading to a long-standing puzzle. Our work resolves this issue by showing that these phenomena originate from Euler-class topology, which survives even when time-reversal symmetry is broken. Moreover, our theory predicts additional topological responses unique to the Euler class, pointing toward new directions for future experiments and materials exploration.

Higher-Order Topological Phases in Spin-Triplet Superconductivity in UTe2

The uranium compound UTe2 has recently attracted broad interest as a promising candidate for spin-triplet superconductivity. Experimental observations such as its extremely large upper critical field, nearly unchanged spin susceptibility below the superconducting transition, and characteristic tunneling spectra all support this possibility. However, a key unresolved question remains: if UTe2 is indeed a spin-triplet superconductor, what kind of topology does it host? This question is highly nontrivial because UTe2 is a multiband system with a complex crystal structure, in sharp contrast to the well-known spin-triplet superfluid helium-3.

In this work, we address this question using a recently developed topological-classification framework for higher-order topological phases. This approach not only determines the topological invariants of a given superconducting state, but also predicts the dimensionality and distribution of the resulting boundary states. We systematically applied this method to all chiral pairing states compatible with the crystal symmetry of UTe2.

Our classification reveals a striking result: except for the trivial spin-singlet pairing state, every symmetry-allowed chiral pairing state in UTe2 realizes a higher-order topological phase. Depending on the pairing symmetry, the system may host third-order topology with Majorana corner states, second-order topology with Majorana hinge states, or even hybrid phases in which corner and hinge states coexist.

These findings suggest that UTe2 provides a fertile platform for realizing and exploring higher-order topological superconductivity in real materials. The predicted boundary states offer experimentally accessible signatures that may help determine the true pairing symmetry of UTe2 and deepen our understanding of spin-triplet superconductivity.

Representation Protected Topology in Spin-Singlet s-Wave Superconductors

Topological superconductors are superconductors that possess nontrivial topological invariants. A hallmark of these systems is the emergence of exotic quasiparticles, known as Majorana quasiparticles, at their boundaries. Because Majorana quasiparticles obey unique quantum statistics, they are considered promising building blocks for topological quantum computation. Early theoretical work predicted that spin-triplet p-wave superconductors naturally host such topological superconducting phases, and this idea has served as a guiding principle in the search for candidate materials. However, spin-triplet p-wave superconductivity is extremely rare in nature, whereas most known superconductors are conventional spin-singlet s-wave superconductors.

In this work, we theoretically demonstrate that topological superconductivity can also emerge in spin-singlet s wave superconductors, in a way that parallels spin triplet systems. The key ingredient is the orbital degree of freedom of electrons, for example p- and d-orbitals, which plays a role analogous to spin when constrained by crystal symmetries. As a result, the orbital structure of the electronic wave functions can give rise to robust topological phases even when the superconducting pairing itself is of the conventional spin-singlet s-wave type.

We further establish a general method to compute the relevant topological invariants in such multiorbital systems. Using this framework, we predict that orbital-driven topological superconducting phases may appear in real materials, including iron-based superconductors, which are among the most intensively studied s-wave multiorbital systems.

These results broaden the landscape of topological superconductivity by showing that unconventional pairing symmetry is not a strict requirement. Instead, orbital structure and symmetry can work together to realize representation protected topology, opening new avenues for discovering topological superconductors in widely studied material families.

Classification of Majorana Multipole Responses under Magnetic Point-Group Symmetry

At the surface of a time reversal symmetric topological superconductor, quasiparticles known as Majorana Kramers pairs necessarily appear. These pairs exhibit highly anisotropic responses to magnetic fields. Earlier theoretical work has shown that such anisotropy gives rise to a rich variety of magnetic responses, collectively referred to as Majorana multipole responses, which are strongly constrained by the crystal symmetry of the superconducting state. Moreover, these responses are closely linked to the Cooper pair symmetry.

However, when time reversal symmetry is broken, Majorana Kramers pairs no longer exist and the usual Majorana multipole response disappears. In multiband superconductors, on the other hand, internal degrees of freedom can cause several Majorana quasiparticles to remain degenerate even without time reversal symmetry. In this case, the magnetic crystal symmetry protects their degeneracy. Motivated by this observation, we extend the theory of Majorana multipole responses to topological superconductors in which time reversal symmetry is broken.

Within this framework, we identify the general symmetry conditions under which Majorana quasiparticles remain degenerate under magnetic point group symmetries, and we show that such degenerate Majorana quasiparticles can exhibit both magnetic and electric multipole responses. We then apply our theory to model systems of two candidate materials: the ferromagnetic superconductor UCoGe and the chiral p-wave superconductor candidate UTe2. In both cases, numerical calculations demonstrate the emergence of a magnetic quadrupole response of the Majorana quasiparticles. This quadrupole response is a characteristic topological phenomenon that appears only when time reversal symmetry is broken.

These results provide a unified way to classify and predict multipole responses of Majorana quasiparticles in superconductors with magnetic crystal symmetry. They also suggest new experimental probes for identifying pairing symmetry and topological character in unconventional superconductors.

Geometric Phases and Surface States in Sublattice Superconductors

In topological superconductors, nontrivial integer valued topological invariants guarantee the appearance of Majorana quasiparticles at the surface. This raises a natural question: what happens when a superconductor instead hosts a nontrivial geometric phase, which takes continuous real values rather than integers? Does the surface still exhibit anomalous behavior?

In this work, we show that the answer is yes. Focusing on superconductors that possess a sublattice degree of freedom, we demonstrate that a nontrivial geometric phase can generate a new type of surface state that is distinct from both Majorana quasiparticles and conventional Andreev bound states. The key feature of this state is that the BCS charge, meaning the difference between electron and hole components of the Bogoliubov quasiparticle wave function, oscillates spatially along the surface. For this reason, we refer to it as an oscillating charge Andreev bound state (OCABS).

OCABS appear only when both the sublattice structure and a finite geometric phase are present. Remarkably, these states carry either purely electron like or purely hole like spectral weight at the surface, in sharp contrast to ordinary Andreev states that mix electrons and holes. As a direct consequence, Andreev reflection is suppressed, leading to striking signatures such as a vanishing tunneling conductance at the corresponding energies.

Our results reveal that geometric phases, usually discussed in the context of band theory, can fundamentally modify superconducting surface physics when internal sublattice structure is present. This opens a pathway to discovering and controlling new classes of surface states beyond the familiar Majorana paradigm, with clear experimental consequences for tunneling spectroscopy and related probes.