Inhomogeneous high temperature melting and decoupling of charge density waves in spin-triplet superconductor UTe2

Charge, spin and Cooper-pair density waves have now been widely detected in exotic superconductors. Understanding how these density waves emerge - and become suppressed by external parameters - is a key research direction in condensed matter physics. Here we study the temperature and magnetic-field evolution of charge density waves in the rare spin-triplet superconductor candidate UTe2 using scanning tunneling microscopy/spectroscopy. We reveal that charge modulations composed of three different wave vectors gradually weaken in a spatially inhomogeneous manner, while persisting to surprisingly high temperatures of 10-12 K. We also reveal an unexpected decoupling of the three-component charge density wave state. Our observations match closely to the temperature scale potentially related to short-range magnetic correlations, providing a possible connection between density waves observed by surface probes and intrinsic bulk features. Importantly, charge density wave modulations become suppressed with magnetic field both below and above superconducting Tc in a comparable manner. Our work points towards an intimate connection between hidden magnetic correlations and the origin of the unusual charge density waves in UTe2.

Migdal-Eliashberg superconductivity in a Kondo lattice

We apply the Migdal-Eliashberg theory of superconductivity to heavy-fermion and mixed valence materials. Specifically, we extend the Anderson lattice model to a case when there exists a strong coupling between itinerant electrons and lattice vibrations. Using the saddle-point approximation, we derive a set of coupled nonlinear equations which describe competition between the crossover to a heavy-fermion or mixed-valence regimes and conventional superconductivity. We find that superconductivity at strong coupling emerges on par with the development of the many-body coherence in a Kondo lattice. Superconductivity is gradually suppressed with the onset of the Kondo screening and for strong electron-phonon coupling the Kondo screening exhibits a characteristic re-entrant behavior. Even though for both weak and strong coupling limits the suppression of superconductivity is weaker in the mixed-valence regime compared to the local moment one, superconducting critical temperature still remains nonzero. In the weak coupling limit the onset of the many body coherence develops gradually, in the strong coupling limit it emerges abruptly in the mixed valence regime while in the local moment regime thef-electrons remain effectively decoupled from the conduction electrons. Possibility of experimental realization of these effects in Ce-based compounds is also discussed.

Orphan high field superconductivity in non-superconducting uranium ditelluride

Reentrant superconductivity is an uncommon phenomenon in which the destructive effects of magnetic field on superconductivity are mitigated, allowing a zero-resistance state to survive under conditions that would otherwise destroy it. Typically, the reentrant superconducting region derives from a zero-field parent superconducting phase. Here, we show that in UTe2 crystals extreme applied magnetic fields give rise to an unprecedented high-field superconductor that lacks a zero-field antecedent. This high-field orphan superconductivity exists at angles offset between 29o and 42o from the crystallographic b to c axes with applied fields between 37 T and 52 T. The stability of field-induced orphan superconductivity presented in this work defies both empirical precedent and theoretical explanation and demonstrates that high-field superconductivity can exist in an otherwise non-superconducting material.

Superconductivity in Quasi-One-Dimensional Ferromagnet CrSbSe3 under High Pressure

Nearly a decade has passed since the discovery of superconductivity in CrAs, but until now, the discovered structure types of chromium-based superconductors are still scanty. It is urgent to expand this family to decipher the interplay between magnetism and superconductivity penetratingly. Here, we report the observation of superconductivity in ferromagnet CrSbSe3 with a quasi-one-dimensional structure under high pressure. Under compression, CrSbSe3 undergoes an insulator-to-metal transition and sequential isostructural phase transitions accompanied by volume collapse. Superconductivity emerges at 32.8 GPa concomitant with metallization in CrSbSe3. A maximum superconducting transition temperature Tc of 7.7 K is achieved at 57.9 GPa benefiting from both the phonon softening and the enhanced p-d hybridization between Se and Cr in CrSbSe3. The discovery of superconductivity in CrSbSe3 expands the existing chromium-based superconductor family and sheds light on the search for concealed superconductivity in low-dimensional van der Waals materials.

Light-induced switching between singlet and triplet superconducting states

While the search for topological triplet-pairing superconductivity has remained a challenge, recent developments in optically stabilizing metastable superconducting states suggest a new route to realizing this elusive phase. Here, we devise a testable theory of competing superconducting orders that permits ultrafast switching to an opposite-parity superconducting phase in centrosymmetric crystals with strong spin-orbit coupling. Using both microscopic and phenomenological models, we show that dynamical inversion symmetry breaking with a tailored light pulse can induce odd-parity (spin triplet) order parameter oscillations in a conventional even-parity (spin singlet) superconductor, which when driven strongly can send the system to a competing minimum in its free energy landscape. Our results provide new guiding principles for engineering unconventional electronic phases using light, suggesting a fundamentally non-equilibrium route toward realizing topological superconductivity.

Discovery of Magnetic Phase Transitions in Heavy-Fermion Superconductor CeRh_{2}As_{2}

We report on the specific heat studies performed on a new generation of CeRh_{2}As_{2} single crystals. Superior quality of the samples and dedicated experimental protocol allowed us to observe an antiferromagneticlike behavior in the normal state and to detect the first-order phase transition of magnetic origin within the superconducting state of the compound. Although in the available literature the physical behavior of CeRh_{2}As_{2} is most often described with the use of quadrupole density wave scenario, we propose an alternative explanation using analogies to antiferromagnetic heavy-fermion superconductors CeRhIn_{5} and Ce_{2}RhIn_{8}.

Resonant Ultrasound Spectroscopy for Irregularly Shaped Samples and Its Application to Uranium Ditelluride

Resonant ultrasound spectroscopy (RUS) is a powerful technique for measuring the full elastic tensor of a given material in a single experiment. Previously, this technique was practically limited to regularly shaped samples such as rectangular parallelepipeds, spheres, and cylinders [W. M. Visscher et al. J. Acoust. Soc. Am. 90, 2154 (1991)JASMAN0001-496610.1121/1.401643]. We demonstrate a new method for determining the elastic moduli of irregularly shaped samples, extending the applicability of RUS to a much larger set of materials. We apply this new approach to the recently discovered unconventional superconductor UTe_{2} and provide its elastic tensor at both 300 and 4 kelvin.

Fully gapped pairing state in spin-triplet superconductor UTe2

The recently discovered superconductor UTe2 is a promising candidate for spin-triplet superconductors, but the symmetry of the superconducting order parameter remains highly controversial. Here, we determine the superconducting gap structure by the thermal conductivity of ultraclean UTe2 single crystals. We find that the a-axis thermal conductivity divided by temperature κ/T in zero-temperature limit is vanishingly small for both magnetic field H‖a and H‖c axes up to H/Hc2 ∼ 0.2, demonstrating the absence of nodes around the a axis contrary to the previous belief. The present results, combined with the reduction of nuclear magnetic resonance Knight shift, indicate that the superconducting order parameter belongs to the isotropic Au representation with a fully gapped pairing state, analogous to the B phase of superfluid 3He. These findings reveal that UTe2 is likely to be a long-sought three-dimensional strong topological superconductor, hosting helical Majorana surface states on any crystal plane.

Two-component nematic superconductivity in 4Hb-TaS2

Most superconductors have an isotropic, single component order parameter and are well described by the standard (BCS) theory for superconductivity. Unconventional, multiple-component superconductors are exceptionally rare and are much less understood. Here, we combine scanning tunneling microscopy and angle-resolved macroscopic transport for studying the candidate chiral superconductor, 4Hb-TaS2. We reveal quasi-periodic one-dimensional modulations in the tunneling conductance accompanied by two-fold symmetric superconducting critical field. The strong modulation of the in-plane critical field, Hc2, points to a nematic, unconventional order parameter. However, the imaged vortex core is isotropic at low temperatures. We suggest a model that reconciles this apparent discrepancy and takes into account previously observed spontaneous time-reversal symmetry breaking at low temperatures. The model describes a competition between a dominating chiral superconducting order parameter and a nematic one. The latter emerges close to the normal phase. Our results strongly support the existence of two-component superconductivity in 4Hb-TaS2 and can provide valuable insights into other systems with coexistent charge order and superconductivity.

Spin-Triplet Superconductivity from Quantum-Geometry-Induced Ferromagnetic Fluctuation

We show that quantum geometry induces ferromagnetic fluctuation resulting in spin-triplet superconductivity. The criterion for ferromagnetic fluctuation is clarified by analyzing contributions from the effective mass and quantum geometry. When the non-Kramers band degeneracy is present near the Fermi surface, the Fubini-Study quantum metric strongly favors ferromagnetic fluctuation. Solving the linearized gap equation with the effective interaction obtained by the random phase approximation, we show that the spin-triplet superconductivity is mediated by quantum-geometry-induced ferromagnetic fluctuation.

Coexistence of topological node surface and Dirac fermions in phonon-mediated superconductor YB2C2

The interaction between nontrivial topology and superconductivity in condensed matter physics has attracted tremendous research interest as it could give rise to exotic phenomena. Herein, based on first-principles calculations, we investigate the electronic structures, mechanical properties, topological properties, dynamic stability, electron-phonon coupling (EPC), and superconducting properties of the synthesized real material YB2C2. It is a tetragonal structure with P4/mbm symmetry and exhibits excellent stability. The calculated electronic band structures reveal that a zero-dimension (0D) Dirac point and two-dimensional (2D) nodal surface coexist near the Fermi level. A spin-orbit coupling (SOC) Dirac point with the topological Fermi arc is observed on the (001) surface. These nodal surfaces are protected by a two-fold screw axis and time-reversal symmetry. Based on the Bardeen-Cooper-Schrieffer theory, the superconducting transition temperature (Tc) in the range 1.25-4.45 K with different Coulomb repulsion constant μ* for YB2C2 is estimated to be consistent with previous experimental results. In addition, the EPC is mainly from the coupling between the dx2-y2 and dz2 orbitals of the Y atom and low-energy phonon modes. The presence of superconductivity and nontrivial topological surface state in YB2C2 suggests that it may be a candidate material for topological superconductors.

Quasi-2D Fermi surface in the anomalous superconductor UTe2

The heavy fermion paramagnet UTe2 exhibits numerous characteristics of spin-triplet superconductivity. Efforts to understand the microscopic details of this exotic superconductivity have been impeded by uncertainty regarding the underlying electronic structure. Here we directly probe the Fermi surface of UTe2 by measuring magnetic quantum oscillations in pristine quality crystals. We find an angular profile of quantum oscillatory frequency and amplitude that is characteristic of a quasi-2D Fermi surface, which we find is well described by two cylindrical Fermi sheets of electron- and hole-type respectively. Additionally, we find that both cylindrical Fermi sheets possess considerable undulation but negligible small-scale corrugation, which may allow for their near-nesting and therefore promote magnetic fluctuations that enhance the triplet pairing mechanism. Importantly, we find no evidence for the presence of any 3D Fermi surface sections. Our results place strong constraints on the possible symmetry of the superconducting order parameter in UTe2.

Field-induced compensation of magnetic exchange as the possible origin of reentrant superconductivity in UTe2

The potential spin-triplet heavy-fermion superconductor UTe2 exhibits signatures of multiple distinct superconducting phases. For field aligned along the b axis, a metamagnetic transition occurs at μ0Hm ≈ 35 T. It is associated with magnetic fluctuations that may be beneficial for the field-reinforced superconductivity surviving up to Hm. Once the field is tilted away from the b towards the c axis, a reentrant superconducting phase emerges just above Hm. In order to better understand this remarkably field-resistant superconducting phase, we conducted magnetic-torque and magnetotransport measurements in pulsed magnetic fields. We determine the record-breaking upper critical field of μ0Hc2 ≈ 73 T and its evolution with angle. Furthermore, the normal-state Hall effect experiences a drastic suppression indicative of a reduced band polarization above Hm in the angular range around 30° caused by a partial compensation between the applied field and an exchange field. This promotes the Jaccarino-Peter effect as a likely mechanism for the reentrant superconductivity above Hm.

Sudden adiabaticity signals reentrant bulk superconductivity in UTe2

There has been a recent surge of interest in UTe2 due to its unconventional magnetic field (H)-reinforced spin-triplet superconducting phases persisting at fields far above the simple Pauli limit for H ∥ [ 010 ] . Magnetic fields in excess of 35 T then induce a field-polarized magnetic state via a first-order-like phase transition. More controversially, for field orientations close to H ∥ [ 011 ] and above 40 T, electrical resistivity measurements suggest that a further superconducting state may exist. However, no Meissner effect or thermodynamic evidence exists to date for this phase making it difficult to exclude alternative scenarios. In this paper, we describe a study using thermal, electrical, and magnetic probes in magnetic fields of up to 55 T applied between the [010] (b) and [001] (c) directions. Our MHz conductivity data reveal the field-induced state of low or vanishing electrical resistance; our simultaneous magnetocaloric effect measurements (i.e. changes in sample temperature due to changing magnetic field) show the first definitive evidence for adiabaticity and thermal behavior characteristic of bulk field-induced superconductivity.

Observation of superconductivity and enhanced upper critical field of η-carbide-type oxide Zr4Pd2O

We report the first observation of bulk superconductivity of a η-carbide-type oxide Zr4Pd2O. The crystal structure and the superconducting properties were studied through synchrotron X-ray diffraction, magnetization, electrical resistivity, and specific heat measurement. The superconducting transition was observed at Tc = 2.73 K. Our measurement revealed that the η-carbide-type oxide superconductor Zr4Pd2O shows an enhanced upper critical field μ0Hc2(0) = 6.72 T, which violates the Pauli-Clogston limit μ0HP = 5.29 T. On the other hand, we found that the enhanced upper critical field is absent in a Rh analogue Zr4Rh2O. The large μ0Hc2(0) of Zr4Pd2O would be raised from strong spin-orbit coupling with Pd-4d electrons. The discovery of new superconducting properties for Zr4Pd2O would shed light on the further development of η-carbide-type oxide superconductors.

Unequivocal Identification of Spin-Triplet and Spin-Singlet Superconductors with Upper Critical Field and Flux Quantization

Spin-triplet superconductors play central roles in Majorana physics and quantum computing but are difficult to identify. We show the methods of kink-point upper critical field and flux quantization in superconducting rings can unequivocally identify spin-singlet, spin-triplet in centrosymmetric superconductors, and singlet-triplet admixture in noncentrosymmetric superconductors, as realized in γ-BiPd, β-Bi_{2}Pd, and α-BiPd, respectively. Our findings are essential for identifying triplet superconductors and exploring their quantum properties.

Longitudinal Spin Fluctuations Driving Field-Reinforced Superconductivity in UTe_{2}

Our measurements of ^{125}Te NMR relaxations reveal an enhancement of electronic spin fluctuations above μ_{0}H^{*}∼15  T, leading to their divergence in the vicinity of the metamagnetic transition at μ_{0}H_{m}≈35  T, below which field-reinforced superconductivity appears when a magnetic field (H) is applied along the crystallographic b axis. The NMR data evidence that these fluctuations are dominantly longitudinal, providing a key to understanding the peculiar superconducting phase diagram in H∥b, where such fluctuations enhance the pairing interactions.

Absence of Spontaneous Magnetic Fields due to Time-Reversal Symmetry Breaking in Bulk Superconducting UTe_{2}

We have investigated the low-temperature local magnetic properties in the bulk of molten salt-flux (MSF)-grown single crystals of the candidate odd-parity superconductor UTe_{2} by zero-field muon spin relaxation (μSR). In contrast to previous μSR studies of UTe_{2} single crystals grown by a chemical vapor transport method, we find no evidence of magnetic clusters or electronic moments fluctuating slow enough to cause a discernible relaxation of the zero-field μSR asymmetry spectrum. Consequently, our measurements on MSF-grown single crystals rule out the generation of spontaneous magnetic fields in the bulk that would occur near impurities or lattice defects if the superconducting state of UTe_{2} breaks time-reversal symmetry. This result suggests that UTe_{2} is characterized by a single-component superconducting order parameter.

Toward large-scale, ordered and tunable Majorana-zero-modes lattice on iron-based superconductors

Majorana excitations are the quasiparticle analog of Majorana fermions in solid materials. Typical examples are the Majorana zero modes (MZMs) and the dispersing Majorana modes. When probed by scanning tunneling spectroscopy, the former manifest as a pronounced conductance peak locating precisely at zero-energy, while the latter behaves as constant or slowly varying density of states. The MZMs obey non-abelian statistics and are believed to be building blocks for topological quantum computing, which is highly immune to the environmental noise. Existing MZM platforms include hybrid structures such as topological insulator, semiconducting nanowire or 1D atomic chains on top of a conventional superconductor, and single materials such as the iron-based superconductors (IBSs) and 4Hb-TaS2. Very recently, ordered and tunable MZM lattice has also been realized in IBS LiFeAs, providing a scalable and applicable platform for future topological quantum computation. In this review, we present an overview of the recent local probe studies on MZMs. Classified by the material platforms, we start with the MZMs in the iron-chalcogenide superconductors where FeTe0.55Se0.45and (Li0.84Fe0.16)OHFeSe will be discussed. We then review the Majorana research in the iron-pnictide superconductors as well as other platforms beyond the IBSs. We further review recent works on ordered and tunable MZM lattice, showing that strain is a feasible tool to tune the topological superconductivity. Finally, we give our summary and perspective on future Majorana research.

A review of UTe2at high magnetic fields

Uranium ditelluride (UTe2) is recognized as a host material to unconventional spin-triplet superconductivity, but it also exhibits a wealth of additional unusual behavior at high magnetic fields. One of the most prominent signatures of the unconventional superconductivity is a large and anisotropic upper critical field that exceeds the paramagnetic limit. This superconductivity survives to 35 T and is bounded by a discontinuous magnetic transition, which itself is also field-direction-dependent. A different, reentrant superconducting phase emerges only on the high-field side of the magnetic transition, in a range of angles between the crystallographicbandcaxes. This review discusses the current state of knowledge of these high-field phases, the high-field behavior of the heavy fermion normal state, and other phases that are stabilized by applied pressure.

Valley-dimensionality locking of superconductivity in cubic phosphides

Two-dimensional superconductivity is primarily realized in atomically thin layers through extreme exfoliation, epitaxial growth, or interfacial gating. Apart from their technical challenges, these approaches lack sufficient control over the Fermiology of superconducting systems. Here, we offer a Fermiology-engineering approach, allowing us to desirably tune the coherence length of Cooper pairs and the dimensionality of superconducting states in arsenic phosphides AsxP1-x under hydrostatic pressure. We demonstrate how this turns these compounds into tunable two-dimensional superconductors with a dome-shaped phase diagram even in the bulk limit. This peculiar behavior is shown to result from an unconventional valley-dimensionality locking mechanism, driven by a delicate competition between three-dimensional hole-type and two-dimensional electron-type energy pockets spatially separated in momentum space. The resulting dimensionality crossover is further discussed to be systematically controllable by pressure and stoichiometry tuning. Our findings pave a unique way to realize and control superconducting phases with special pairing and dimensional orders.

Exchange interaction for the triplet superconductor UTe2

UTe2 is one triplet superconductor that defies conventional relation between ferromagnetism and superconductivity. Our search for a theoretical explanation starts with one spin-triplet state of two electrons and construct a two-particle exchange interaction that favors the formation of Cooper pairs under the configuration. A modified application of the Bardeen-Cooper-Schrieffer (BCS) theory using parameters derived from ab-initio density functional calculations for electrons and phonons enables us to derive the critical temperature of 1.64 K and an average superconducting gap of 0.25 meV at 0 K. We extend the investigation further into the superconductivity under pressure, showing how Tc and the gap of UTe2 change under compression in ways that are consistent with the results of experiment.

Topological Superconductors from a Materials Perspective

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

Superconducting spin reorientation in spin-triplet multiple superconducting phases of UTe2

Superconducting (SC) state has spin and orbital degrees of freedom, and spin-triplet superconductivity shows multiple SC phases because of the presence of these degrees of freedom. However, the observation of spin-direction rotation occurring inside the SC state (SC spin rotation) has hardly been reported. Uranium ditelluride, a recently found topological superconductor, exhibits various SC phases under pressure: SC state at ambient pressure (SC1), high-temperature SC state above 0.5 gigapascal (SC2), and low-temperature SC state above 0.5 gigapascal (SC3). We performed nuclear magnetic resonance (NMR) and ac susceptibility measurements on a single-crystal uranium ditelluride. The b axis spin susceptibility remains unchanged in SC2, unlike in SC1, and decreases below the SC2-SC3 transition with spin modulation. These unique properties in SC3 arise from the coexistence of two SC order parameters. Our NMR results confirm spin-triplet superconductivity with SC spin parallel to b axis in SC2 and unveil the remaining of spin degrees of freedom in SC uranium ditelluride.

Revealing a 3D Fermi Surface Pocket and Electron-Hole Tunneling in UTe_{2} with Quantum Oscillations

Spin triplet superconductor UTe_{2} is widely believed to host a quasi-two-dimensional Fermi surface, revealed by first-principles calculations, photoemission, and quantum oscillation measurements. An outstanding question still remains as to the existence of a three-dimensional Fermi surface pocket, which is crucial for our understanding of the exotic superconducting and topological properties of UTe_{2}. This 3D Fermi surface pocket appears in various theoretical models with different physics origins, but has not been unambiguously detected in experiment. Here for the first time we provide concrete evidence for a relatively isotropic, small Fermi surface pocket of UTe_{2} via quantum oscillation measurements. In addition, we observed high frequency quantum oscillations corresponding to electron-hole tunneling between adjacent electron and hole pockets. The coexistence of 2D and 3D Fermi surface pockets, as well as the breakdown orbits, provide a test bed for theoretical models and aid the realization of a unified understanding of the superconducting state of UTe_{2} from the first-principles approach.

Superconductivity beyond the Conventional Pauli Limit in High-Pressure CeSb_{2}

We report the discovery of superconductivity at a pressure-induced magnetic quantum phase transition in the Kondo lattice system CeSb_{2}, sustained up to magnetic fields that exceed the conventional Pauli limit eightfold. Like CeRh_{2}As_{2}, CeSb_{2} is locally noncentrosymmetric around the Ce site, but the evolution of critical fields and normal state properties as CeSb_{2} is tuned through the quantum phase transition motivates a fundamentally different explanation for its resilience to applied field.

Monolayer Superconductivity and Tunable Topological Electronic Structure at the Fe(Te,Se)/Bi2 Te3 Interface

The interface between 2D topological Dirac states and an s-wave superconductor is expected to support Majorana-bound states (MBS) that can be used for quantum computing applications. Realizing these novel states of matter and their applications requires control over superconductivity and spin-orbit coupling to achieve spin-momentum-locked topological interface states (TIS) which are simultaneously superconducting. While signatures of MBS have been observed in the magnetic vortex cores of bulk FeTe_0.55 Se_0.45 , inhomogeneity and disorder from doping make these signatures unclear and inconsistent between vortices. Here superconductivity is reported in monolayer (ML) FeTe_1-y Se_y (Fe(Te,Se)) grown on Bi₂ Te₃ by molecular beam epitaxy (MBE). Spin and angle-resolved photoemission spectroscopy (SARPES) directly resolve the interfacial spin and electronic structure of Fe(Te,Se)/Bi₂ Te₃ heterostructures. For y = 0.25, the Fe(Te,Se) electronic structure is found to overlap with the Bi₂ Te₃ TIS and the desired spin-momentum locking is not observed. In contrast, for y = 0.1, reduced inhomogeneity measured by scanning tunneling microscopy (STM) and a smaller Fe(Te,Se) Fermi surface with clear spin-momentum locking in the topological states are found. Hence, it is demonstrated that the Fe(Te,Se)/Bi₂ Te₃ system is a highly tunable platform for realizing MBS where reduced doping can improve characteristics important for Majorana interrogation and potential applications.

Detection of a pair density wave state in UTe2

Spin-triplet topological superconductors should exhibit many unprecedented electronic properties, including fractionalized electronic states relevant to quantum information processing. Although UTe₂ may embody such bulk topological superconductivity^1-11, its superconductive order parameter Δ(k) remains unknown¹². Many diverse forms for Δ(k) are physically possible¹² in such heavy fermion materials¹³. Moreover, intertwined^14,15 density waves of spin (SDW), charge (CDW) and pair (PDW) may interpose, with the latter exhibiting spatially modulating^14,15 superconductive order parameter Δ(r), electron-pair density^16-19 and pairing energy gap^17,20-23. Hence, the newly discovered CDW state²⁴ in UTe₂ motivates the prospect that a PDW state may exist in this material^24,25. To search for it, we visualize the pairing energy gap with μeV-scale energy resolution using superconductive scanning tunnelling microscopy (STM) tips^26-31. We detect three PDWs, each with peak-to-peak gap modulations of around 10 μeV and at incommensurate wavevectors P_i=1,2,3 that are indistinguishable from the wavevectors Q_i=1,2,3 of the prevenient²⁴ CDW. Concurrent visualization of the UTe₂ superconductive PDWs and the non-superconductive CDWs shows that every P_i:Q_i pair exhibits a relative spatial phase δϕ ≈ π. From these observations, and given UTe₂ as a spin-triplet superconductor¹², this PDW state should be a spin-triplet PDW^24,25. Although such states do exist³² in superfluid ³He, for superconductors, they are unprecedented.

Magnetic-field-sensitive charge density waves in the superconductor UTe2

The intense interest in triplet superconductivity partly stems from theoretical predictions of exotic excitations such as non-Abelian Majorana modes, chiral supercurrents and half-quantum vortices^1-4. However, fundamentally new and unexpected states may emerge when triplet superconductivity appears in a strongly correlated system. Here we use scanning tunnelling microscopy to reveal an unusual charge-density-wave (CDW) order in the heavy-fermion triplet superconductor UTe₂ (refs. ^5-8). Our high-resolution maps reveal a multi-component incommensurate CDW whose intensity gets weaker with increasing field, with the CDW eventually disappearing at the superconducting critical field H_c2. To understand the phenomenology of this unusual CDW, we construct a Ginzburg-Landau theory for a uniform triplet superconductor coexisting with three triplet pair-density-wave states. This theory gives rise to daughter CDWs that would be sensitive to magnetic field owing to their origin in a pair-density-wave state and provides a possible explanation for our data. Our discovery of a CDW state that is sensitive to magnetic fields and strongly intertwined with superconductivity provides important information for understanding the order parameters of UTe₂.

Modulations in Superconductors: Probes of Underlying Physics

The importance of modulations is elevated to an unprecedented level, due to the delicate conditions required to bring out exotic phenomena in quantum materials, such as topological materials, magnetic materials, and superconductors. Recently, state-of-the-art modulation techniques in material science, such as electric-double-layer transistor, piezoelectric-based strain apparatus, angle twisting, and nanofabrication, have been utilized in superconductors. They not only efficiently increase the tuning capability to the broader ranges but also extend the tuning dimensionality to unprecedented degrees of freedom, including quantum fluctuations of competing phases, electronic correlation, and phase coherence essential to global superconductivity. Here, for a comprehensive review, these techniques together with the established modulation methods, such as elemental substitution, annealing, and polarization-induced gating, are contextualized. Depending on the mechanism of each method, the modulations are categorized into stoichiometric manipulation, electrostatic gating, mechanical modulation, and geometrical design. Their recent advances are highlighted by applications in newly discovered superconductors, e.g., nickelates, Kagome metals, and magic-angle graphene. Overall, the review is to provide systematic modulations in emergent superconductors and serve as the coordinate for future investigations, which can stimulate researchers in superconductivity and other fields to perform various modulations toward a thorough understanding of quantum materials.

Orbital Fulde-Ferrell-Larkin-Ovchinnikov state in an Ising superconductor

In superconductors possessing both time and inversion symmetries, the Zeeman effect of an external magnetic field can break the time-reversal symmetry, forming a conventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state characterized by Cooper pairings with finite momentum^1,2. In superconductors lacking (local) inversion symmetry, the Zeeman effect may still act as the underlying mechanism of FFLO states by interacting with spin-orbit coupling (SOC). Specifically, the interplay between the Zeeman effect and Rashba SOC can lead to the formation of more accessible Rashba FFLO states that cover broader regions in the phase diagram^3-5. However, when the Zeeman effect is suppressed because of spin locking in the presence of Ising-type SOC, the conventional FFLO scenarios are no longer effective. Instead, an unconventional FFLO state is formed by coupling the orbital effect of magnetic fields with SOC, providing an alternative mechanism in superconductors with broken inversion symmetries^6-8. Here we report the discovery of such an orbital FFLO state in the multilayer Ising superconductor 2H-NbSe₂. Transport measurements show that the translational and rotational symmetries are broken in the orbital FFLO state, providing the hallmark signatures of finite-momentum Cooper pairings. We establish the entire orbital FFLO phase diagram, consisting of a normal metal, a uniform Ising superconducting phase and a six-fold orbital FFLO state. This study highlights an alternative route to achieving finite-momentum superconductivity and provides a universal mechanism to preparing orbital FFLO states in similar materials with broken inversion symmetries.

Chiral superconductivity in UTe2 probed by anisotropic low-energy excitations

Chiral spin-triplet superconductivity is a topologically nontrivial pairing state with broken time-reversal symmetry, which can host Majorana quasiparticles. The heavy-fermion superconductor UTe₂ exhibits peculiar properties of spin-triplet pairing, and the possible chiral state has been actively discussed. However, the symmetry and nodal structure of its order parameter in the bulk, which determine the Majorana surface states, remains controversial. Here we focus on the number and positions of superconducting gap nodes in the ground state of UTe₂. Our magnetic penetration depth measurements for three field orientations in three crystals all show the power-law temperature dependence with exponents close to 2, which excludes single-component spin-triplet states. The anisotropy of low-energy quasiparticle excitations indicates multiple point nodes near the k_y- and k_z-axes in momentum space. These results can be consistently explained by a chiral B_3u + iA_u non-unitary state, providing fundamentals of the topological properties in UTe₂.

Field Induced Multiple Superconducting Phases in UTe_{2} along Hard Magnetic Axis

The superconducting (SC) phase diagram in uranium ditelluride is explored under magnetic fields (H) along the hard magnetic b axis using a high-quality single crystal with T_{c}=2.1  K. Simultaneous electrical resistivity and ac magnetic susceptibility measurements discern low- and high-field SC (LFSC and HFSC, respectively) phases with contrasting field-angular dependence. Crystal quality increases the upper critical field of the LFSC phase, but the H^{*} of ∼15  T, at which the HFSC phase appears, is always the same through the various crystals. A phase boundary signature is also observed inside the LFSC phase near H^{*}, indicating an intermediate SC phase characterized by small flux pinning forces.

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}.

Signature of spin-triplet exciton condensations in LaCoO3 at ultrahigh magnetic fields up to 600 T

Bose-Einstein condensation of electron-hole pairs, exciton condensation, has been effortfully investigated since predicted 60 years ago. Irrefutable evidence has still been lacking due to experimental difficulties in verifying the condensation of the charge neutral and non-magnetic spin-singlet excitons. Whilst, condensation of spin-triplet excitons is a promising frontier because spin supercurrent and spin-Seebeck effects will be observable. A canonical cobaltite LaCoO₃ under very high magnetic fields is a propitious candidate, yet to be verified. Here, we unveil the exotic phase diagram of LaCoO₃ up to 600 T generated using the electromagnetic flux compression method and the state-of-the-art magnetostriction gauge. We found the continuous magnetostriction curves and a bending structure, which suggest the emergence of two distinct spin-triplet exciton condensates. By constructing a phenomenological model, we showed that quantum fluctuations of excitons are crucial for the field-induced successive transitions. The spin-triplet exciton condensation in a cobaltite, which is three-dimensional and thermally equilibrated, opens up a novel venue for spintronics technologies with spin-supercurrent such as a spin Josephson junction.

Triplet Pairing Mechanisms from Hund's-Kondo Models: Applications to UTe_{2} and CeRh_{2}As_{2}

Observing that several U and Ce based candidate triplet superconductors share a common structural motif, with pairs of magnetic atoms separated by an inversion center, we hypothesize a triplet pairing mechanism based on an interplay of Hund's and Kondo interactions that is unique to this structure. In the presence of Hund's interactions, valence fluctuations generate a triplet superexchange between electrons and local moments. The offset from the center of symmetry allows spin-triplet pairs formed by the resulting Kondo effect to delocalize onto the Fermi surface, precipitating superconductivity. We demonstrate this mechanism within a minimal two-channel Kondo lattice model and present support for this pairing mechanism from existing experiments.

Long-range skin Josephson supercurrent across a van der Waals ferromagnet

The emerging field of superconducting spintronics promises new quantum device architectures without energy dissipation. When entering a ferromagnet, a supercurrent commonly behaves as a spin singlet that decays rapidly; in contrast, a spin-triplet supercurrent can transport over much longer distances, and is therefore more desirable, but so far has been observed much less frequently. Here, by using the van der Waals ferromagnet Fe3GeTe2 (F) and spin-singlet superconductor NbSe2 (S), we construct lateral Josephson junctions of S/F/S with accurate interface control to realize long-range skin supercurrent. The observed supercurrent across the ferromagnet can extend over 300 nm, and exhibits distinct quantum interference patterns in an external magnetic field. Strikingly, the supercurrent displays pronounced skin characteristics, with its density peaked at the surfaces or edges of the ferromagnet. Our central findings shed new light on the convergence of superconductivity and spintronics based on two-dimensional materials.

Itinerant-localized dichotomy in magnetic anisotropic properties of U-based ferromagnets

The electronic structure, spin and orbital magnetic moments, and the magnetic anisotropy energy in selected U-based compounds are investigated making use of the correlated band theory. First, we demonstrate that the LSDA+U approach with exact atomic limit implemented as a combination of the relativistic density functional theory with the Anderson impurity model provides a good quantitative description for UGa[Formula: see text]. Further, the method is applied to UFe[Formula: see text] and UFe[Formula: see text]Si[Formula: see text] ferromagnets. The calculated positive uniaxial magnetic anisotropy together with negative enthalpy of formation for UFe[Formula: see text]Si[Formula: see text] make it as a candidate for the magnetically hard materials. Our studies suggest a viable route for further development of the rare-earth-lean permanent magnets by replacing a part of U atoms by some rare-earth like Sm in UFe[Formula: see text]Si[Formula: see text].

Topological Structure of the Order Parameter of Unconventional Superconductors Based on d- and f- Elements

The superconducting order parameter (SOP) of a triplet superconductor UTe2 was constructed using the topological space group approach, in which, in contrast to phenomenological and topological approaches, the single pair function and phase winding in condensate are different quantities. The connection between them is investigated for the D2h point group and the m′m′m magnetic group. It is shown how a non-unitary pair function of UTe2 can be constructed using one-dimensional real irreducible representations and Ginzburg–Landau phase winding. It is also shown that the total phase winding is non-zero in magnetic symmetry only. Experimental data on the superconducting order parameter of topological superconductors UPt3, Sr2RuO4, LaPt3P, and UTe2 are considered and peculiarities of their nodal structures are connected with the theoretical results of the topological space group approach.

An emerging global picture of heavy fermion physics

Recent progresses using state-of-the-art experimental techniques have motivated a number of new insights on heavy fermion physics. This article gives a brief summary of the author's research along this direction. We discuss five major topics including: (1) development of phase coherence and two-stage hybridization; (2) two-fluid behavior and hidden universal scaling; (3) quantum phase transitions and fractionalized heavy fermion liquid; (4) quantum critical superconductivity; (5) material-specific properties. These cover the most essential parts of heavy fermion physics and lead to an emerging global picture beyond conventional theories based on mean-field or local approximations.

Topologically Localized Insulators

We show that fully localized, three-dimensional, time-reversal-symmetry-broken insulators do not belong to a single phase of matter but can realize topologically distinct phases that are labeled by integers. The phase transition occurs only when the system becomes conducting at some filling. We find that these novel topological phases are fundamentally distinct from insulators without disorder: they are guaranteed to host delocalized boundary states giving rise to the quantized boundary Hall conductance, whose value is equal to the bulk topological invariant.

Surface Cooper-Pair Spin Waves in Triplet Superconductors

We study the electrodynamics of spin triplet superconductors including dipolar interactions, which give rise to an interplay between the collective spin dynamics of the condensate and orbital Meissner screening currents. Within this theory, we identify a class of spin waves that originate from the coupled dynamics of the spin-symmetry breaking triplet order parameter and the electromagnetic field. In particular, we study magnetostatic spin wave modes that are localized to the sample surface. We show that these surface modes can be excited and detected using experimental techniques such as microwave spin wave resonance spectroscopy or nitrogen-vacancy magnetometry, and propose that the detection of these modes offers a means for the identification of spin triplet superconductivity.

Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires

In most naturally occurring superconductors, electrons with opposite spins form Cooper pairs. This includes both conventional s-wave superconductors such as aluminium, as well as high-transition-temperature, d-wave superconductors. Materials with intrinsic p-wave superconductivity, hosting Cooper pairs made of equal-spin electrons, have not been conclusively identified, nor synthesized, despite promising progress^1-3. Instead, engineered platforms where s-wave superconductors are brought into contact with magnetic materials have shown convincing signatures of equal-spin pairing^4-6. Here we directly measure equal-spin pairing between spin-polarized quantum dots. This pairing is proximity-induced from an s-wave superconductor into a semiconducting nanowire with strong spin-orbit interaction. We demonstrate such pairing by showing that breaking a Cooper pair can result in two electrons with equal spin polarization. Our results demonstrate controllable detection of singlet and triplet pairing between the quantum dots. Achieving such triplet pairing in a sequence of quantum dots will be required for realizing an artificial Kitaev chain^7-9.

Advances in actinide thin films: synthesis, properties, and future directions

Actinide-based compounds exhibit unique physics due to the presence of 5f electrons, and serve in many cases as important technological materials. Targeted thin film synthesis of actinide materials has been successful in generating high-purity specimens in which to study individual physical phenomena. These films have enabled the study of the unique electron configuration, strong mass renormalization, and nuclear decay in actinide metals and compounds. The growth of these films, as well as their thermophysical, magnetic, and topological properties, have been studied in a range of chemistries, albeit far fewer than most classes of thin film systems. This relative scarcity is the result of limited source material availability and safety constraints associated with the handling of radioactive materials. Here, we review recent work on the synthesis and characterization of actinide-based thin films in detail, describing both synthesis methods and modeling techniques for these materials. We review reports on pyrometallurgical, solution-based, and vapor deposition methods. We highlight the current state-of-the-art in order to construct a path forward to higher quality actinide thin films and heterostructure devices.

Observation of Coexisting Dirac Bands and Moiré Flat Bands in Magic-Angle Twisted Trilayer Graphene

Moiré superlattices that consist of two or more layers of 2D materials stacked together with a small twist angle have emerged as a tunable platform to realize various correlated and topological phases, such as Mott insulators, unconventional superconductivity, and quantum anomalous Hall effect. Recently, magic-angle twisted trilayer graphene (MATTG) has shown both robust superconductivity similar to magic-angle twisted bilayer graphene and other unique properties, including the Pauli-limit violating and re-entrant superconductivity. These rich properties are deeply rooted in its electronic structure under the influence of distinct moiré potential and mirror symmetry. Here, combining nanometer-scale spatially resolved angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy, the as-yet unexplored band structure of MATTG near charge neutrality is systematically measured. These measurements reveal the coexistence of the distinct dispersive Dirac band with the emergent moiré flat band, showing nice agreement with the theoretical calculations. These results serve as a stepstone for further understanding of the unconventional superconductivity in MATTG.

Electronic landscape of the f-electron intermetallics with the ThCr2Si2 structure

Although strongly correlated f-electron systems are well known as reservoirs for quantum phenomena, a persistent challenge is to design specific states. What is often missing are simple ways to determine whether a given compound can be expected to exhibit certain behaviors and what tuning vector(s) would be useful to select the ground state. In this review, we address this question by aggregating information about Ce, Eu, Yb, and U compounds with the ThCr₂Si₂ structure. We construct electronic/magnetic state maps that are parameterized in terms of unit cell volumes and d-shell filling, which reveals useful trends including that (i) the magnetic and nonmagnetic examples are well separated, and (ii) the crossover regions harbor the examples with exotic states. These insights are used to propose structural/chemical regions of interest in these and related materials, with the goal of accelerating discovery of the next generation of f-electron quantum materials.

High-Throughput Investigations of Topological and Nodal Superconductors

The theory of symmetry indicators has enabled database searches for topological materials in normal conducting phases, which has led to several encyclopedic topological material databases. To date, such a database for topological superconductors is yet to be achieved because of the lack of information about pairing symmetries of realistic materials. In this Letter, sidestepping this issue, we tackle an alternative problem: the predictions of topological and nodal superconductivity in materials for each single-valued representation of point groups. Based on recently developed symmetry indicators for superconductors, we provide comprehensive mappings from pairing symmetries to the topological or nodal superconducting nature for nonmagnetic materials listed in the Inorganic Crystal Structure Database. We quantitatively show that around 90% of computed materials are topological or nodal superconductors when a pairing that belongs to a one-dimensional nontrivial representation of point groups is assumed. When materials are representation-enforced nodal superconductors, positions and shapes of the nodes are also identified. When combined with experiments, our results will help us understand the pairing mechanism and facilitate realizations of the long-sought Majorana fermions promising for topological quantum computations.

Investigating the A n+1B n X3n+1 Homologous Series: A New Platform for Studying Magnetic Praseodymium Based Intermetallics

The recent discovery of the A _n+1B _n X_3n+1 (A = lanthanide, B = transition metal, X = tetrel) homologous series provides a new platform to study the structure-property relationships of highly correlated electron systems. Several members of Ce _n+1Co _n Ge_3n+1 (n = 1, 4, 5, 6, and ∞) show evidence of heavy electron behavior with complex magnetic interactions. While the Ce analogues have been investigated, only n = 1, 2, and ∞ of Pr _n+1Co _n Ge_3n+1 have been synthesized, with n = 1 and 2 showing a nonsinglet magnetic ground state. The Pr analogues can provide a platform for direct comparison of highly correlated behavior. In this perspective, we discuss the impetus for synthesizing the Pr _n+1Co _n Ge_3n+1 members and present the structural characterization of the n = 3 and n = 4 members. We lay the foundation for future investigations of the Pr _n+1Co _n Ge_3n+1 family of compounds and highlight the importance of complementary methods to characterize new quantum materials.

Pressured-induced superconducting phase with large upper critical field and concomitant enhancement of antiferromagnetic transition in EuTe2

We report an unusual pressure-induced superconducting state that coexists with an antiferromagnetic ordering of Eu²⁺ moments and shows a large upper critical field comparable to the Pauli paramagnetic limit in EuTe₂. In concomitant with the emergence of superconductivity with T_c ≈ 3–5 K above P_c ≈ 6 GPa, the antiferromagnetic transition temperature T_N(P) experiences a quicker rise with the slope increased dramatically from dT_N/dP = 0.85(14) K/GPa for P ≤ P_c to 3.7(2) K/GPa for P ≥ P_c. Moreover, the superconducting state can survive in the spin-flop state with a net ferromagnetic component of the Eu²⁺ sublattice under moderate magnetic fields μ₀H ≥ 2 T. Our findings establish the pressurized EuTe₂ as a rare magnetic superconductor possessing an intimated interplay between magnetism and superconductivity.

Here, the authors report pressure-induced superconductivity with concomitant enhancement of antiferromagnetic transition in layered EuTe₂. The superconductivity is distinctly characterized by the high upper critical fields exceeding the Pauli limit among binary tellurides, a prerequisite of the coexistence of ferromagnetism with superconductivity.

Triplet Superconductivity from Nonlocal Coulomb Repulsion in an Atomic Sn Layer Deposited onto a Si(111) Substrate

Atomic layers deposited on semiconductor substrates introduce a platform for the realization of the extended electronic Hubbard model, where the consideration of electronic repulsion beyond the on-site term is paramount. Recently, the onset of superconductivity at 4.7 K has been reported in the hole-doped triangular lattice of tin atoms on a silicon substrate. Through renormalization group methods designed for weak and intermediate coupling, we investigate the nature of the superconducting instability in hole-doped Sn/Si(111). We find that the extended Hubbard nature of interactions is crucial to yield triplet pairing, which is f-wave (p-wave) for moderate (higher) hole doping. In light of persisting challenges to tailor triplet pairing in an electronic material, our finding promises to pave unprecedented ways for engineering unconventional triplet superconductivity.