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Quantum phase transitions arise in many-body systems because of competing interactions that promote rivaling ground states. Recent years have seen the identification of continuous quantum phase transitions, or quantum critical points, in a host of antiferromagnetic heavy-fermion compounds. Studies of the interplay between the various effects have revealed new classes of quantum critical points and are uncovering a plethora of new quantum phases. At the same time, quantum criticality has provided fresh insights into the electronic, magnetic, and superconducting properties of the heavy-fermion metals. We review these developments, discuss the open issues, and outline some directions for future research.

Nontrivial topology in condensed-matter systems enriches quantum states of matter to go beyond either the classification into metals and insulators in terms of conventional band theory or that of symmetry-broken phases by Landau’s order parameter framework. So far, focus has been on weakly interacting systems, and little is known about the limit of strong electron correlations. Heavy fermion systems are a highly versatile platform to explore this regime. Here we report the discovery of a giant spontaneous Hall effect in the Kondo semimetal Ce₃Bi₄Pd₃ that is noncentrosymmetric but preserves time-reversal symmetry. We attribute this finding to Weyl nodes—singularities of the Berry curvature—that emerge in the immediate vicinity of the Fermi level due to the Kondo interaction. We stress that this phenomenon is distinct from the previously detected anomalous Hall effect in materials with broken time-reversal symmetry; instead, it manifests an extreme topological response that requires a beyond-perturbation-theory description of the previously proposed nonlinear Hall effect. The large magnitude of the effect in even tiny electric and zero magnetic fields as well as its robust bulk nature may aid the exploitation in topological quantum devices.

Flat electronic bands are expected to show proportionally enhanced electron correlations, which may generate a plethora of novel quantum phases and unusual low-energy excitations. They are increasingly being pursued in d -electron-based systems with crystalline lattices that feature destructive electronic interference, where they are often topological. Such flat bands, though, are generically located far away from the Fermi energy, which limits their capacity to partake in the low-energy physics. Here we show that electron correlations produce emergent flat bands that are pinned to the Fermi energy. We demonstrate this effect within a Hubbard model, in the regime described by Wannier orbitals where an effective Kondo description arises through orbital-selective Mott correlations. Moreover, the correlation effect cooperates with symmetry constraints to produce a topological Kondo semimetal. Our results motivate a novel design principle for Weyl Kondo semimetals in a new setting, viz. d -electron-based materials on suitable crystal lattices, and uncover interconnections among seemingly disparate systems that may inspire fresh understandings and realizations of correlated topological effects in quantum materials and beyond. Flat electronic bands can give rise to correlation-driven phases but for this, they need to be tuned to the Fermi level. Here the authors predict flat bands pinned at the Fermi level due to orbital-selective interactions and discuss implications for the design of topological Kondo semimetal in d-electron systems.

Superconductivity originates from the formation of bound (Cooper) pairs of electrons that can move through the lattice without resistance below the superconducting transition temperature T c (ref.  1 ). Electron Cooper pairs in most superconductors form anti-parallel spin singlets with total spin S  = 0 (ref.  2 ), although they can also form parallel spin-triplet Cooper pairs with S  = 1 and an odd parity wavefunction 3 . Spin-triplet pairing is important because it can host topological states and Majorana fermions relevant for quantum computation 4 , 5 . Because spin-triplet pairing is usually mediated by ferromagnetic (FM) spin fluctuations 3 , uranium-based materials near an FM instability are considered to be ideal candidates for realizing spin-triplet superconductivity 6 . Indeed, UTe 2 , which has a T c  ≈ 1.6 K (refs.  7 , 8 ), has been identified as a candidate for a chiral spin-triplet topological superconductor near an FM instability 7 – 14 , although it also has antiferromagnetic (AF) spin fluctuations 15 , 16 . Here we use inelastic neutron scattering (INS) to show that superconductivity in UTe 2 is coupled to a sharp magnetic excitation, termed resonance 17 – 23 , at the Brillouin zone boundary near AF order. Because the resonance has only been found in spin-singlet unconventional superconductors near an AF instability 17 – 23 , its observation in UTe 2 suggests that AF spin fluctuations may also induce spin-triplet pairing 24 or that electron pairing in UTe 2 has a spin-singlet component. Inelastic neutron scattering measurements show that superconductivity in UTe 2 is associated with a resonance near antiferromagnetic order that suggests an unexpected spin-singlet component to the electron pairing.

Recent experiments in multiband Fe-based and heavy-fermion superconductors have challenged the long-held dichotomy between simple s- and d-wave spin-singlet pairing states. Here, we advance several time-reversal-invariant irreducible pairings that go beyond the standard singlet functions through a matrix structure in the band/orbital space, and elucidate their naturalness in multiband systems. We consider the sτ3 multiorbital superconducting state for Fe-chalcogenide superconductors. This state, corresponding to a d + d intra- and inter-band pairing, is shown to contrast with the more familiar d + id state in a way analogous to how the B- triplet pairing phase of 3He superfluid differs from its A- phase counterpart. In addition, we construct an analog of the sτ3 pairing for the heavy-fermion superconductor CeCu2Si2, using degrees-of-freedom that incorporate spin-orbit coupling. Our results lead to the proposition that d-wave superconductors in correlated multiband systems will generically have a fully-gapped Fermi surface when they are examined at sufficiently low energies.

An important challenge in condensed matter physics is understanding iron-based superconductors. Among these systems, the iron selenides hold the record for highest superconducting transition temperature and pose especially striking puzzles regarding the nature of superconductivity. The pairing state of the alkaline iron selenides appears to be of d -wave type based on the observation of a resonance mode in neutron scattering, while it seems to be of s -wave type from the nodeless gaps observed everywhere on the Fermi surface. Here we propose an orbital-selective pairing state, dubbed sτ 3 , as a natural explanation of these disparate properties. The pairing function, containing a matrix τ 3 in the basis of 3 d -electron orbitals, does not commute with the kinetic part of the Hamiltonian. This dictates the existence of both intraband and interband pairing terms in the band basis. A spin resonance arises from a d -wave-type sign change in the intraband pairing component, whereas the quasiparticle excitation is fully gapped on the FS due to an s -wave-like form factor associated with the addition in quadrature of the intraband and interband pairing terms. We demonstrate that this pairing state is energetically favored when the electron correlation effects are orbitally selective. More generally, our results illustrate how the multiband nature of correlated electrons affords unusual types of superconducting states, thereby shedding new light not only on the iron-based materials but also on a broad range of other unconventional superconductors such as heavy fermion and organic systems. Unconventional superconductivity: Orbital selective pairing in iron selenides Orbital-selective pairing could explain the unusual properties observed in the unconventional superconductor iron selenide. Conventional superconductivity arises when electrons form Cooper pairs due to electron-phonon coupling. In some materials, however, unconventional superconductivity can arise, which is driven by electron-electron rather than electron-phonon couplings. The detailed mechanism that facilitates electron pairing in unconventional systems remains elusive but iron selenide systems could help to provide insights as they exhibit both relatively high temperature superconductivity, and also strong electron correlations. With different experiments suggesting different pairing mechanisms, however, these systems are somewhat puzzling. An international team of researchers led by Qimiao Si from Rice University now theoretically demonstrate that an orbital-selective pairing state could explain this unusual behaviour, which may also be at play in other unconventional superconductors such as heavy fermion and organic systems.

Understanding the microscopic origins of electronic phases in high-transition temperature (high-Tc) superconductors is important for elucidating the mechanism of superconductivity. In the paramagnetic tetragonal phase of BaFe2–xTxAs2 (where T is Co or Ni) iron pnictides, an in-plane resistivity anisotropy has been observed. Here, we use inelastic neutron scattering to show that low-energy spin excitations in these materials change from fourfold symmetric to twofold symmetric at temperatures corresponding to the onset of the in-plane resistivity anisotropy. Because resistivity and spin excitation anisotropies both vanish near optimal superconductivity, we conclude that they are likely intimately connected.

Strong correlations in matter promote a landscape of quantum phases and associated quantum critical points. For metallic systems, there is increasing recognition that the quantum criticality goes beyond the Landau framework and, thus, further means are needed to characterize the quantum critical fluid. Here we do so by studying an entanglement quantity, the quantum Fisher information, in a strange metal system, focusing on the exemplary case of an Anderson/Kondo lattice model near its Kondo destruction quantum critical point. The spin quantum Fisher information peaks at the quantum critical point and indicates a strongly entangled ground state. Our results are supported by the quantum Fisher information extracted from inelastic neutron scattering measurements in heavy fermion metals. Our work elucidates the loss of quasiparticles in strange metals, opens a quantum information avenue to advance the understanding of metallic quantum criticality in a broad range of strongly correlated systems, and points to a regime of quantum matter to realize amplified entanglement. Strange metals exhibit anomalous electronic and thermodynamic properties near a quantum critical point. Here the authors show theoretically that entanglement quantified by quantum Fisher information is amplified in a strange metal at a quantum critical point, which is supported by analysis of experimental data.

The origin of the electronic nematicity in FeSe is one of the most important unresolved puzzles in the study of iron-based superconductors. In both spin- and orbital-nematic models, the intrinsic magnetic excitations at Q 1  = (1, 0) and Q 2  = (0, 1) of twin-free FeSe are expected to provide decisive criteria for clarifying this issue. Although a spin-fluctuation anisotropy below 10 meV between Q 1 and Q 2 has been observed by inelastic neutron scattering at low temperature, it remains unclear whether such an anisotropy also persists at higher energies and associates with the nematic transition T s . Here we use resonant inelastic X-ray scattering to probe the high-energy magnetic excitations of detwinned FeSe. A prominent anisotropy between the magnetic excitations along the H and K directions is found to persist to E  ≈ 200 meV, which decreases gradually with increasing temperature and finally vanishes at a temperature around T s . The measured high-energy spin excitations are dispersive and underdamped, which can be understood from a local-moment perspective.Taking together the large energy scale far beyond the d x z / d y z orbital splitting, we suggest that the nematicity in FeSe is probably spin-driven. The mechanism that drives nematic behaviour in iron-based superconductors is still unclear. Now, nematicity and anisotropy in spin excitations are shown to disappear at the same temperature, indicating that the transition is primarily spin-driven.

Quantum phase transitions (QPTs) occur at zero temperature when parameters such as magnetic field or pressure are varied. In heavy fermion compounds, superconductivity often accompanies QPTs, a seeming exception being the material YbRh2Si2, which undergoes a magnetic QPT. Schuberth et al. performed magnetic and calorimetric measurements at extremely low temperatures and magnetic fields and found that it does become superconducting after all. Almost simultaneously with superconductivity, another order appeared that showed signatures of nuclear spin origin. Science, this issue p. 485 The smooth disappearance of antiferromagnetic order in strongly correlated metals commonly furnishes the development of unconventional superconductivity. The canonical heavy-electron compound YbRh2Si2 seems to represent an apparent exception from this quantum critical paradigm in that it is not a superconductor at temperature T ≥ 10 millikelvin (mK). Here we report magnetic and calorimetric measurements on YbRh2Si2, down to temperatures as low as T [approximate] 1 mK. The data reveal the development of nuclear antiferromagnetic order slightly above 2 mK and of heavy-electron superconductivity almost concomitantly with this order. Our results demonstrate that superconductivity in the vicinity of quantum criticality is a general phenomenon.