On the Heavy Fermion Road

Quantum-well states at the surface of a heavy-fermion superconductor

Two-dimensional electronic states at surfaces are often observed in simple wide-band metals such as Cu or Ag (refs. 1-4). Confinement by closed geometries at the nanometre scale, such as surface terraces, leads to quantized energy levels formed from the surface band, in stark contrast to the continuous energy dependence of bulk electron bands2,5-10. Their energy-level separation is typically hundreds of meV (refs. 3,6,11). In a distinct class of materials, strong electronic correlations lead to so-called heavy fermions with a strongly reduced bandwidth and exotic bulk ground states12,13. Quantum-well states in two-dimensional heavy fermions (2DHFs) remain, however, notoriously difficult to observe because of their tiny energy separation. Here we use millikelvin scanning tunnelling microscopy (STM) to study atomically flat terraces on U-terminated surfaces of the heavy-fermion superconductor URu2Si2, which exhibits a mysterious hidden-order (HO) state below 17.5 K (ref. 14). We observe 2DHFs made of 5f electrons with an effective mass 17 times the free electron mass. The 2DHFs form quantized states separated by a fraction of a meV and their level width is set by the interaction with correlated bulk states. Edge states on steps between terraces appear along one of the two in-plane directions, suggesting electronic symmetry breaking at the surface. Our results propose a new route to realize quantum-well states in strongly correlated quantum materials and to explore how these connect to the electronic environment.

Unconventional superconductivity in UTe2

The novel spin-triplet superconductor candidate UTe₂was discovered only recently at the end of 2018 and already attracted enormous attention. We review key experimental and theoretical progress which has been achieved in different laboratories. UTe₂is a heavy-fermion paramagnet, but following the discovery of superconductivity, it has been expected to be close to a ferromagnetic instability, showing many similarities to the U-based ferromagnetic superconductors, URhGe and UCoGe. This view might be too simplistic. The competition between different types of magnetic interactions and the duality between the local and itinerant character of the 5fUranium electrons, as well as the shift of the U valence appear as key parameters in the rich phase diagrams discovered recently under extreme conditions like low temperature, high magnetic field, and pressure. We discuss macroscopic and microscopic experiments at low temperature to clarify the normal phase properties at ambient pressure for field applied along the three axis of this orthorhombic structure. Special attention will be given to the occurrence of a metamagnetic transition atH_m= 35 T for a magnetic field applied along the hard magnetic axisb. Adding external pressure leads to strong changes in the magnetic and electronic properties with a direct feedback on superconductivity. Attention is paid on the possible evolution of the Fermi surface as a function of magnetic field and pressure. Superconductivity in UTe₂is extremely rich, exhibiting various unconventional behaviors which will be highlighted. It shows an exceptionally huge superconducting upper critical field with a re-entrant behavior under magnetic field and the occurrence of multiple superconducting phases in the temperature-field-pressure phase diagrams. There is evidence for spin-triplet pairing. Experimental indications exist for chiral superconductivity and spontaneous time reversal symmetry breaking in the superconducting state. Different theoretical approaches will be described. Notably we discuss that UTe₂is a possible example for the realization of a fascinating topological superconductor. Exploring superconductivity in UTe₂reemphasizes that U-based heavy fermion compounds give unique examples to study and understand the strong interplay between the normal and superconducting properties in strongly correlated electron systems.

Unveiling hidden multipolar orders with magnetostriction

Broken symmetries in solids involving higher order multipolar degrees of freedom are historically referred to as “hidden orders” due to the formidable task of detecting them with conventional probes. In this work, we theoretically propose that magnetostriction provides a powerful and novel tool to directly detect higher-order multipolar symmetry breaking—such as the elusive octupolar order—by examining scaling behaviour of length change with respect to an applied magnetic field h. Employing a symmetry-based Landau theory, we focus on the family of Pr-based cage compounds with strongly correlated f-electrons, Pr(Ti,V,Ir)₂(Al,Zn)₂₀, whose low energy degrees of freedom are purely higher-order multipoles: quadrupoles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{O}}_{20,22}$$\end{document}O20,22 and octupole \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{T}}_{xyz}$$\end{document}Txyz. We demonstrate that a magnetic field along the [111] direction induces a distinct linear-in-h length change below the octupolar ordering temperature. The resulting “magnetostriction coefficient” is directly proportional to the octupolar order parameter, thus providing clear access to such subtle order parameters.

Higher-order multipolar phases are unusual states that can form in correlated materials and are difficult to observe as they do not directly couple to conventional probes. Patri et al. theoretically show that angle-dependent magnetostriction measurements can probe quadrupolar and octupolar ordering.

Tomonaga-Luttinger liquid behavior and spinon confinement in YbAlO3

Low dimensional quantum magnets are interesting because of the emerging collective behavior arising from strong quantum fluctuations. The one-dimensional (1D) S = 1/2 Heisenberg antiferromagnet is a paradigmatic example, whose low-energy excitations, known as spinons, carry fractional spin S = 1/2. These fractional modes can be reconfined by the application of a staggered magnetic field. Even though considerable progress has been made in the theoretical understanding of such magnets, experimental realizations of this low-dimensional physics are relatively rare. This is particularly true for rare-earth-based magnets because of the large effective spin anisotropy induced by the combination of strong spin-orbit coupling and crystal field splitting. Here, we demonstrate that the rare-earth perovskite YbAlO3 provides a realization of a quantum spin S = 1/2 chain material exhibiting both quantum critical Tomonaga-Luttinger liquid behavior and spinon confinement-deconfinement transitions in different regions of magnetic field-temperature phase diagram.

Grüneisen parameter studies on heavy fermion quantum criticality

The Grüneisen parameter, experimentally determined from the ratio of thermal expansion to specific heat, quantifies the pressure dependence of characteristic energy scales of matter. It is highly enhanced for Kondo lattice systems, whose properties are strongly dependent on the pressure sensitive antiferromagnetic exchange interaction between f- and conduction electrons. In this review, we focus on the divergence of the Grüneisen parameter and its magnetic analogue, the adiabatic magnetocaloric effect, for heavy-fermion metals near quantum critical points. We compare experimental results with current theoretical models, including the effect of strong geometrical frustration. We also discuss the possibility of using materials with the divergent magnetic Grüneisen parameter for adiabatic demagnetization cooling to very low temperatures.

Superconductivity-induced re-entrance of the orthorhombic distortion in Ba1-xKxFe2As2

Detailed knowledge of the phase diagram and the nature of the competing magnetic and superconducting phases is imperative for a deeper understanding of the physics of iron-based superconductivity. Magnetism in the iron-based superconductors is usually a stripe-type spin-density-wave, which breaks the tetragonal symmetry of the lattice, and is known to compete strongly with superconductivity. Recently, it was found that in some systems an additional spin-density-wave transition occurs, which restores this tetragonal symmetry, however, its interaction with superconductivity remains unclear. Here, using thermodynamic measurements on Ba_1−xK_xFe₂As₂ single crystals, we show that the spin-density-wave phase of tetragonal symmetry competes much stronger with superconductivity than the stripe-type spin-density-wave phase, which results in a novel re-entrance of the latter at or slightly below the superconducting transition.

The interplay between magnetic and superconducting phases is important to understand the physics of iron-based superconductivity. Here, the authors use thermodynamic measurements on Ba_1−xK_xFe₂As₂ single crystals to provide details of its phase diagram and the re-entrance of a C₂ spin-density-wave phase.

Evidence for Coexistence of Bulk Superconductivity and Itinerant Antiferromagnetism in the Heavy Fermion System CeCo(In(1-x)Cdx)5

In the generic phase diagram of heavy fermion systems, tuning an external parameter such as hydrostatic or chemical pressure modifies the superconducting transition temperature. The superconducting phase forms a dome in the temperature—tuning parameter phase diagram, which is associated with a maximum of the superconducting pairing interaction. Proximity to antiferromagnetism suggests a relation between the disappearance of antiferromagnetic order and superconductivity. We combine muon spin rotation, neutron scattering, and x-ray absorption spectroscopy techniques to gain access to the magnetic and electronic structure of CeCo(In_1−xCd_x)₅ at different time scales. Different magnetic structures are obtained that indicate a magnetic order of itinerant character, coexisting with bulk superconductivity. The suppression of the antiferromagnetic order appears to be driven by a modification of the bandwidth/carrier concentration, implying that the electronic structure and consequently the interplay of superconductivity and magnetism is strongly affected by hydrostatic and chemical pressure.

Heavy fermions. Unconventional Fermi surface in an insulating state

Insulators occur in more than one guise; a recent finding was a class of topological insulators, which host a conducting surface juxtaposed with an insulating bulk. Here, we report the observation of an unusual insulating state with an electrically insulating bulk that simultaneously yields bulk quantum oscillations with characteristics of an unconventional Fermi liquid. We present quantum oscillation measurements of magnetic torque in high-purity single crystals of the Kondo insulator SmB6, which reveal quantum oscillation frequencies characteristic of a large three-dimensional conduction electron Fermi surface similar to the metallic rare earth hexaborides such as PrB6 and LaB6. The quantum oscillation amplitude strongly increases at low temperatures, appearing strikingly at variance with conventional metallic behavior.

Temperature dependent tunneling spectroscopy in the heavy fermion CeRu2Si2 and in the antiferromagnet CeRh2Si2

CeRu(2)Si(2) and CeRh(2)Si(2) are two similar heavy fermion stoichiometric compounds located on the two sides of a magnetic quantum critical phase transition. CeRh(2)Si(2) is an antiferromagnet below T(N) = 36 K with moderate electronic masses whereas CeRu(2)Si(2) is a paramagnetic metal with particularly heavy electrons. Here we present tunneling spectroscopy measurements as a function of temperature (from 0.15 to 45 K). The tunneling conductance at 0.15 K reveals V-shaped dips around the Fermi level in both compounds, which disappear in CeRu(2)Si(2) above the coherence temperature, and in CeRh(2)Si(2) above the Néel temperature. In the latter case, two different kinds of V-shaped tunneling conductance dips are found.

Thermoelectric response near a quantum critical point of β-YbAlB4 and YbRh2Si2: a comparative study

The thermoelectric coefficients have been measured down to a very low temperature for the Yb-based heavy-fermion compounds β-YbAlB4 and YbRh2Si2, often considered as model systems for the local quantum criticality case. We observe a striking difference in the behavior of the Seebeck coefficient S in the vicinity of their respective quantum critical point (QCP). Approaching the critical field, S/T is enhanced in β-YbAlB4, but drastically reduced in YbRh2Si2. The ratio of thermopower to specific heat remains constant for β-YbAlB4, but it is significantly reduced near the QCP in YbRh2Si2. In both systems, on the other hand, the Nernst coefficient shows a diverging behavior near the QCP. The interplay between valence and magnetic quantum criticality and the additional possibility of a Lifshitz transition crossing the critical field under magnetic field are discussed as the origin of the different behaviors of these compounds.

Thermal expansion and Grüneisen parameters of Ba(Fe(1-x)Co(x))2As2: a thermodynamic quest for quantum criticality

Thermal expansion data are used to study the uniaxial pressure dependence of the electronic-magnetic entropy of Ba(Fe(1-x)Co(x))2As2. Uniaxial pressure is found to be proportional to doping and, thus, also an appropriate tuning parameter in this system. Many of the features predicted to occur for a pressure-tuned quantum critical system, in which superconductivity is an emergent phase hiding the critical point, are observed. The electronic-magnetic Grüneisen parameters associated with the spin-density wave and superconducting transitions further demonstrate an intimate connection between both ordering phenomena.

Magnetic structure of CeRhIn(5) under magnetic field

The magnetically ordered ground state of CeRhIn(5) at ambient pressure and zero magnetic field is an incommensurate helicoidal phase with the propagation vector k = (1/2,1/2,0.298) and the magnetic moment in the basal plane of the tetragonal structure. We determined by neutron diffraction the two different magnetically ordered phases of CeRhIn(5) evidenced by bulk measurements under applied magnetic field in the basal plane. The low temperature high magnetic phase corresponds to a commensurate sine-wave structure of the magnetization with k = (1/2,1/2,1/4). At high temperature, the phase is incommensurate with k = (1/2,1/2,0.298) and a possible small ellipticity. The propagation vector of this phase is the same as that of the zero-field structure.