The identification, control and theoretical modelling of spin-triplet superconductors (STC) remain a central theme in quantum materials research. Intrinsic STC are rare but offer rich condensate properties and unique surface properties allowing insights into the nature of the spin-triplet order, and promising applications in quantum technologies. Owing to interactions, the order parameter in STCs can often be intertwined with other symmetry breaking orders like charge/spin density waves (CDW/SDW) or pair density waves (PDW) complicating their phase diagrams. UTe2 stands out as the only known odd-parity, STC that harbors such intertwined orders on the surface and possible topological surface states composed of Majorana fermions. While the (0-11) facet is the most heavily studied, the fermiology of this surface that gives rise to such exotic phenomena is still lacking and continues to be an area of active interest. Here, we employ low-temperature spectroscopic imaging to reveal the Fermi surface of UTe2 through quasiparticle interference. We find scattering originating from the uranium-derived bands that play a major role in the formation of the CDW and the PDW phases. Tunneling spectroscopy further reveals spectral signatures of the CDW gap, corroborating its onset temperature. Suppressing the CDW with a magnetic field, highlights the presence of small, circular Fermi pockets that disperse strongly near the Fermi energy. We discuss the nature of the interference patterns and the origin of the small Fermi pockets in the context of the calculated band structure and the unconventional CDW phase.

We perform a theoretical investigation of the nature of vortices in chiral d+id superconductors on the kagome lattice. The study is motivated by recent experimental developments reporting evidence of time-reversal symmetry breaking in the superconducting state of kagome metals. Using self-consistent microscopic calculations that incorporate the characteristics of the band structure of the kagome lattice, we find that fractional vortices permeate the ground state condensate in the presence of an external field. Each fractional vortex carries one third of the superconducting flux quantum and exhibits a characteristic signature related to one of the three sublattice degrees of freedom of the kagome lattice. We discuss the relevance of these results to recent experimental studies of kagome superconductors in the presence of an external magnetic field.

Unconventional superconductivity is a desirable state of matter due to its potential for high transition temperatures T_c and associated favorable superconducting properties. However, the sign-changing nature of the order parameter of unconventional superconductors renders their condensates fragile to disorder, an inevitability in real materials. We uncover the generic properties of electronic band structures and associated Bloch weights able to support robust unconventional superconductivity. We demonstrate this property in several case studies of the kagome and Lieb lattices, showing how unconventional superconductors exhibit unusually weak T_c suppression by disorder, despite featuring fully compensated sign-changing order parameters. We contrast these results with those for unconventional superconductivity on the square and honeycomb lattices, which are unable to protect the condensates from disorder. Finally, we discuss material candidates for which this effect may be realized.

Subgap bound states can produce pronounced features in the density of states without necessarily giving rise to a corresponding signal in ultrasonic attenuation. We show that the coherence factors act as a filter for the contribution of bound states to ultrasonic attenuation. In a phase-biased superconductor–normal–superconductor junction, the Andreev bound states produce a resonance in the attenuation when the phonon frequency matches the phase-dependent separation between the subgap levels. The situation is different at a (110) grain boundary in a d_x^2-y^2 superconductor. There, nonmagnetic impurity-induced bound states remain weak in the attenuation despite their large spectral weight, because the relevant coherence factors nearly cancel. When local magnetism develops near the boundary, this cancellation is removed, the bound states become spin split, and a clear attenuation peak appears, which we argue is governed by a process involving emission of phonons by pair of quasiparticles at low temperatures. We obtain these results using a real-space Bogoliubov–de Gennes formulation of ultrasonic attenuation and benchmark the method against the known low-temperature behavior of homogeneous s- and d-wave superconductors. The results show that ultrasound can distinguish bound states with similar spectral signatures but different symmetry and spin structure.

Since the prediction of a time-reversal symmetry breaking (TRSB) d + id' state in twisted bilayer cuprate superconductors by Can et al. [Nat. Phys. 17, 519 (2021)], several experiments have attempted to detect this state, yielding conflicting results. At present, it is not clear which differences in samples or experimental conditions might explain these discrepancies. In this work, we perform a tight-binding lattice model calculation with phenomenological interlayer tunneling, examining the order parameter as a function of twist angle, interlayer tunneling, doping, and temperature. We observe the TRSB state to be correlated to the position of the Van Hove singularity in the normal state, which changes not only as a function of doping but also the tunneling strength. Two such phases are identified as nominally consistent with in-plane d + id' and d + is order, but with unexpected transformation properties under bilayer symmetry operations. We calculate the Josephson critical current, in particular examining the angle dependence for various tunneling strengths. Finally, we discuss the existing experiments in the context of our results.

Intrinsic topological superconductors with protected boundary modes obeying non-Abelian statistics constitute a vanishingly small class of quantum materials. A defining spectroscopic signature of such phases is the presence of in-gap topological surface states (TSS). However, despite extensive theoretical proposals, their unambiguous experimental identification has remained elusive. Here we use vector magnetic-field scanning tunnelling microscopy to obtain direct spectroscopic evidence of TSS in the spin-triplet superconductor UTe2. Atomic-scale spectroscopy reveals striking site-dependent superconductivity: Te sites host a large in-gap density of states that nearly fills the superconducting gap, whereas neighboring atomic sites remain gapped. Upon application of a magnetic field, the in-gap states on the Te sites are selectively suppressed, yielding a spatially homogeneous superconducting state with a markedly deeper gap relative to zero field. This site-selective gap evolution is in quantitative agreement with theoretical predictions for TSS in UTe2 that possess dominant Te-orbital character. Spectral-function calculations incorporating the Zeeman coupling reproduce the observed magnetic-field response. Our results provide a spectroscopic fingerprint of the long-sought TSS in superconductors and establish UTe2 as a compelling system for exploring intrinsic topological superconductivity.

Odd-parity magnetism constitutes an intriguing phase of matter which breaks inversion symmetry while preserving time-reversal symmetry. Here we demonstrate that the Fe-based superconductors exhibiting coplanar magnetic order realize an odd-parity magnetic state by combining low-energy modeling with density-functional theory. In the absence of spin-orbit coupling, the electronic spins are polarized along the $k_z$-direction and the splitting of the up and down states exhibits an $h$-wave form-factor. The magnitude of the splitting depends sensitively on specific parameters of the low-energy model, including specific out-of-plane hopping parameters and the Fermi energies of the hole- and electron-pockets. Interestingly, despite this state breaking inversion symmetry and exhibiting a finite out-of-plane Berry curvature and non-linear anomalous Hall effect, the Edelstein effect vanishes. Incorporating spin-orbit coupling tilts the momentum-space electronic spins into the ($k_x,k_y$)-plane and imparts finite in-plane components to the Edelstein response. Our findings highlight the Fe-based superconductors as platforms for exploring odd-parity magnetism both on its own and coexisting with unconventional superconductivity.

A variety of experimental observations in monolayer transition metal dichalcogenide superconductors with Ising spin-orbit coupling suggest the presence of an unconventional superconducting pairing mechanism. Some of these experiments include observation of Leggett modes and a nodal superconducting gap in STM experiments, a large in-plane upper critical field compared to the Pauli limit, and the observation of a two-fold gap anisotropy in magnetoresistance measurements. Here, we propose a superconducting pairing mechanism mediated by spin and charge fluctuations and identify the dominant superconducting instability relevant to monolayer TaS_2. We then explore the effect of an additional electron-phonon pairing contribution, and compare our results with recent experimental findings. In particular, our theory stabilizes a superconducting ground state with nodal-like density of states that agrees with STM experiments. The theory obtains a large in-plane upper critical field due to a combination of Ising spin-orbit coupling and even-odd parity mixing in the superconducting state. Further, we find that an in-plane magnetic field splits the degeneracy of the superconducting ground state, and the resulting two-fold symmetric superconducting order parameter could explain the gap anisotropy observed in magnetoresistance experiments. Overall, the proposed theoretical pairing model can reconcile diverse experimental observations and remains consistent with observations on other dichalcogenide superconductors such as monolayer NbSe_2.

Altermagnetic metals break time-reversal symmetry and feature spin-split Fermi surfaces generated by compensated N & eacute;el-ordered collinear magnetic moments. Being metallic, such altermagnets may undergo a further instability at low temperatures to a superconducting state, and it is an interesting open question what the salient features are of such altermagnetic superconductors. We address this question on the basis of realistic microscopic models that capture the altermagnetic sublattice degree of freedom. We find that the sublattice structure can strongly affect the superconducting gap structure in altermagnetic superconductors. In particular, it imposes nodes in the gap on the Brillouin zone edges for superconductors stabilized by momentum-independent bare attraction channels. We contrast this to the case of superconductivity generated by extended range interactions where pairing is allowed on the Brillouin zone edges and both spin-singlet and equal-spin-pairing triplet states can be stabilized. Equal-spin-pairing triplet superconductivity is generically favored in the limit of large altermagnetic spin splitting of the bands compared to the superconducting gap scale, and features characteristic nonunitary properties arising from the altermagnetic order.

The inverse Lieb lattice has recently emerged as a promising platform for altermagnetism, with several materials with this structure proposed as d-wave altermagnetic candidates. Here, we develop a symmetry-based microscopic Hamiltonian for these materials that includes both sublattice and orbital degrees of freedom, going beyond the sublattice-only minimal models that have been extensively used to study such altermagnets. We apply these models to examine multi-orbital electron correlation physics in the vanadium oxychalcogenide family altermagnets, which contain dominant xy and xz/yz orbitals character at the Fermi level in the altermagnetic state. We demonstrate that xy orbitals are crucial to stabilize the altermagnetic state observed within a single V_2O layer, and altermagnetic order in the xz/yz orbitals is induced through Hund's coupling. Additionally, we show that these multi-orbital models reveal topological regimes in which topological edge states are naturally orbital selective.

We analyze the spin susceptibility of spin-triplet superconductors from the zero-field to finite-field regimes, with emphasis on its implications for Knight-shift measurements. In the zero-field limit, we review the general expression for the static spin susceptibility and highlight the universal zero-temperature sum rule, ∑_i χ_ii(T=0)=2χ^N, which constrains the residual susceptibility components for any triplet state. Using representative isotropic, helical, and chiral d⃗-vectors, we illustrate how the Knight shift encodes the spin configuration of the order parameter and show that the sum rule remains robust even for anisotropic Fermi surfaces. We then incorporate magnetic field effects through a semiclassical Doppler shift of quasiparticle energies in the vortex state. The resulting field dependence of the susceptibility - including both longitudinal (Knight-shift) and transverse magnetic susceptibility components - provides a sensitive probe of nodal directions and the momentum dependence of the d⃗-vector. Applying this framework to UTe_2, we demonstrate how the distinct irreducible representations allowed by orthorhombic symmetry can be differentiated by their field-dependent susceptibility.

The iron-based superconductor FeTe0.55Se0.45 is a peculiar material: it hosts surface states with a Dirac dispersion, is a putative topological superconductor hosting Majorana modes in vortices, and has an unusually low Fermi energy. The superconducting state is generally characterized by three gaps in different bands, with the homogeneous, spatially extended Bogoliubov excitations─in this work, we uncover evidence that it is instead of a very different nature. Our scanning tunneling spectroscopy data show several peaks in the density of states above a full gap, and by analyzing their spatial and junction-resistance dependence, we conclude the peaks above the first one are not coherence peaks from different bands. Instead, comparisons with our simulations indicate they originate from generalized Shiba states that are spatially overlapping. This can lead to an amorphous state of Bogoliubov quasiparticles, reminiscent of impurity bands in semiconductors. We discuss the origin and implications of this new state.

Odd-parity magnetism constitutes an intriguing phase of matter which breaks inversion symmetry while preserving time-reversal symmetry. Here we demonstrate that the Fe-based superconductors exhibiting coplanar magnetic order realize an odd-parity magnetic state by combining low-energy modeling with density-functional theory. In the absence of spin-orbit coupling, the electronic spins are polarized along the k_z-direction and the splitting of the up and down states exhibits an h-wave form-factor. The magnitude of the splitting depends sensitively on specific parameters of the low-energy model, including specific out-of-plane hopping parameters and the Fermi energies of the hole- and electron-pockets. Interestingly, despite this state breaking inversion symmetry and exhibiting a finite out-of-plane Berry curvature and non-linear anomalous Hall effect, the Edelstein effect vanishes. Incorporating spin-orbit coupling tilts the momentum-space electronic spins into the (k_x,k_y)-plane and imparts finite in-plane components to the Edelstein response. Our findings highlight the Fe-based superconductors as platforms for exploring odd-parity magnetism both on its own and coexisting with unconventional superconductivity.

The presence of a non-centrosymmetric crystal structure and in-plane mirror symmetry allows an Ising spin-orbit coupling to form in some two-dimensional materials. Examples include transition metal dichalcogenide superconductors like monolayer NbSe_2, MoS_2, TaS_2, and PbTe_2, where a nontrivial nature of the superconducting state is currently being explored. In this study, we develop a microscopic formalism for Ising superconductors that captures the superconducting instability arising from a momentum-dependent spin- and charge-fluctuation-mediated pairing interaction. We apply our pairing model to the electronic structure of monolayer NbSe_2, where first-principles calculations reveal the presence of strong paramagnetic fluctuations. Our calculations provide a quantitative measure of the mixing between the even- and odd-parity superconducting states and its variation with Coulomb interaction. Further, numerical analysis in the presence of an external Zeeman field reveals the role of Ising spin-orbit coupling and mixing of odd-parity superconducting state in influencing the low-temperature enhancement of the critical magnetic field.

The heavy-fermion compound UTe_2 is a candidate for hosting intrinsic spin-triplet superconductivity. At present, however, the type of triplet Cooper pairing realized in UTe_2 remains unknown, which calls for further experimental and theoretical investigations. In this paper, we develop a microscopic minimal model for the superconducting phases of UTe_2 based on recent findings in the description of its low-energy normal state electronic properties. We apply the resulting theoretical model to extract the nodal gap properties of the allowed superconducting ground states, and determine their associated topological surface states on the experimentally relevant (0-11) cleave plane. We find that the Fermi surface of UTe_2 enforces additional point nodes in excess to the point nodes imposed by symmetry, which may reconcile several experiments seemingly in conflict with B_2u or B_3u pairing symmetries. Furthermore, we map out the in-gap Majorana surface-bound modes existing on the (0-11) surface, and discuss their potential for additional insight into the pairing structure of UTe_2.

Realizing odd-parity, time-reversal-preserving, nonrelativistic spin splitting is a central goal for spintronics applications. We propose a group-theory-based microscopic framework to induce odd-parity spin splitting from coplanar antiferromagnetic (AFM) states without spin-orbit coupling (SOC). We develop phenomenological models for 421 conventional period-doubling AFM systems in nonsymmorphic space groups and construct minimal microscopic models for 119 of these. We find that these AFM states can attain three possible competing ground states. These ground states all break symmetries in addition to those broken by the usual AFM order. Specifically, they give rise to either odd-parity spin-splitting, nematic order, or scalar odd-parity order related to multiferroicity. Our microscopic theories reveal that the odd-parity spin-splitting energy scale is generically large and further reveal that the scalar odd-parity order gives a nonzero Berry curvature dipole without SOC. We identify 67 materials in the Magndata database for which our theory applies. We provide density-functional theory (DFT) calculations on Fe-based materials that reveal an h-wave spin splitting consistent with our symmetry arguments and apply our microscopic model to determine the nonrelativistic Edelstein response for CeNiAsO.

Several recent experiments have challenged the premise that cuprate high-temperature superconductors approach conventional Landau-BCS behavior in the high-doping limit. We argue, based on an analysis of their superconducting spectra, that anomalous properties seen in the most-studied overdoped cuprates require a pairing interaction that is strongly inhomogeneous on nm length scales. This is consistent with recent proposals that the “strange-metal” phase above T_{c} in the same doping range arises from a spatially random interaction. We show, via mean-field Bogoliubov-de Gennes (BdG) calculations and time-dependent Ginzburg-Landau (TDGL) simulations, that key features of the observed tunneling spectra are reproduced when both inhomogeneity and thermal phase fluctuations are accounted for. In accord with experiments, BdG calculations find that low-T spectra are highly inhomogeneous and exhibit a low-energy spectral shoulder and broad coherence peaks. However, the spectral gap in this approach becomes homogeneous at high T, in contrast to experiments. This is resolved when thermal fluctuations are included within TDGL; in this case, global phase coherence is lost at the superconducting T_{c} via a broadened BKT transition, while robust phase-coherent superconducting islands persist well above T_{c}. The local spectrum remains inhomogeneous at T_{c}, and the gap is found to fill instead of close with increasing temperature.

Quasiparticle interference (QPI) obtained from scanning tunneling microscopy (STM) is a powerful method to help extract the pairing symmetry of unconventional superconductors. We examine the general properties of QPI on surfaces of spin-triplet superconductors. It is shown how the multicomponent nature of the spin-triplet d[over →]-vector and the general existence of topological surface states in triplet condensates both offer important differences from QPI on spin-singlet superconductors. We then turn to a microscopic model relevant for the spin-triplet candidate UTe_{2} and compare the computed QPI with recent STM measurements. We conclude that the two candidate pairing instabilities B_{2u} and B_{3u} exhibit distinct features in the QPI intensity which allows for distinguishing them using the experimental data. Characteristic features of the emergent topological surface states protected by chiral symmetry in general, and by mirror symmetries in the case of UTe_{2}, provide further unique signatures to help pinpoint the pairing symmetry channel of this material.

Altermagnets constitute a class of collinear compensated N & eacute;el ordered magnets that break time-reversal symmetry and feature spin-split band structures. Based on versatile microscopic models able to capture the altermagnetic sublattice degrees of freedom, we study characteristic local signatures of altermagnetism near disorder sites. We give a complete list of two-dimensional models that exhibit altermagnetism classified by their corresponding layer groups. Specifically, we calculate the local density of states in the vicinity of pointlike nonmagnetic impurities and expose its spatial dependence for two minimal models showcasing d-wave and g-wave altermagnetism. The momentum structure of the nodes (d-wave, g-wave, etc.) is directly imprinted on the total local density of states, thus measurable by scanning tunneling conductance experiments. This signature is present both in the spin-resolved as well as the spin-summed local density of states. We find a weaker response in the nonmagnetic state from the anisotropic crystal environment and uncover the importance of the sublattice degree of freedom to model altermagnets. We also study coexistence phases of altermagnetism and superconductivity and provide predictions for the local impurity response of in-gap bound states. The response of impurity bound states strongly enhances the distinct altermagnetic signature.

Altermagnetism defies conventional classifications of collinear magnetic phases, standing apart from ferromagnetism and antiferromagnetism with its unique combination of spin-dependent symmetries, net-zero magnetization, and anomalous Hall transport. Although altermagnetic states have been realized experimentally, their integration into functional devices has been hindered by the structural rigidity and poor tunability of existing materials. First, through cobalt intercalation of the superconducting 2H-NbSe_2 polymorph, we induce and stabilize a robust altermagnetic phase and using both theory and experiment, we directly observe the lifting of Kramers degeneracy. Additionally, we present spectroscopic insight into a previously hinted low-temperature phase, and provide evidence of its electronic origin. While shedding light on overlooked aspects of altermagnetism, these findings open pathways to spin-based technologies and lay a foundation for advancing the emerging field of altertronics.