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 atHm= 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.

Fermi-Surface Instability in the Heavy-Fermion Superconductor UTe_{2}.

Transport measurements are presented up to fields of 29 T in the recently discovered heavy-fermion superconductor UTe_{2} with magnetic field H applied along the easy magnetization a axis of the body-centered orthorhombic structure. The thermoelectric power varies linearly with temperature above the superconducting transition, T_{SC}=1.5 K, indicating that superconductivity develops in a Fermi liquid regime. As a function of field the thermoelectric power shows successive anomalies which appear at critical values of the magnetic polarization. Remarkably, the lowest magnetic field instability for H∥a occurs for the same critical value of the magnetization (0.4 µ_{B}) than the first order metamagnetic transition at 35 T for field applied along the b axis. It can be clearly identified as a Lifshitz transition. The estimated number of charge carriers at low temperature reveals a metallic ground state distinct from LDA calculations indicating that strong electronic correlations are a major issue.

Field-induced spin-density wave beyond hidden order in URu2Si2.

URu2Si2 is one of the most enigmatic strongly correlated electron systems and offers a fertile testing ground for new concepts in condensed matter science. In spite of >30 years of intense research, no consensus on the order parameter of its low-temperature hidden-order phase exists. A strong magnetic field transforms the hidden order into magnetically ordered phases, whose order parameter has also been defying experimental observation. Here, thanks to neutron diffraction under pulsed magnetic fields up to 40 T, we identify the field-induced phases of URu2Si2 as a spin-density-wave state. The transition to the spin-density wave represents a unique touchstone for understanding the hidden-order phase. An intimate relationship between this magnetic structure, the magnetic fluctuations and the Fermi surface is emphasized, calling for dedicated band-structure calculations.

Collapse of Ferromagnetism and Fermi Surface Instability near Reentrant Superconductivity of URhGe.

We present thermoelectric power and resistivity measurements in the ferromagnetic superconductor URhGe for a magnetic field applied along the hard magnetization b axis of the orthorhombic crystal. Reentrant superconductivity is observed near the spin reorientation transition at H_{R}=12.75 T, where a first order transition from the ferromagnetic to the polarized paramagnetic state occurs. Special focus is given to the longitudinal configuration, where both the electric and heat current are parallel to the applied field. The validity of the Fermi-liquid T^{2} dependence of the resistivity through H_{R} demonstrates clearly that no quantum critical point occurs at H_{R}. Thus, the ferromagnetic transition line at H_{R} becomes first order implying the existence of a tricritical point at finite temperature. The enhancement of magnetic fluctuations in the vicinity of the tricritical point stimulates the reentrance of superconductivity. The abrupt sign change observed in the thermoelectric power with the thermal gradient applied along the b axis together with the strong anomalies in the other directions is definitive macroscopic evidence that in addition a significant change of the Fermi surface appears through H_{R}.

Thermal Conductivity through the Quantum Critical Point in YbRh_{2}Si_{2} at Very Low Temperature.

The thermal conductivity of YbRh_{2}Si_{2} has been measured down to very low temperatures under field in the basal plane. An additional channel for heat transport appears below 30 mK, both in the antiferromagnetic and paramagnetic states, respectively, below and above the critical field suppressing the magnetic order. This excludes antiferromagnetic magnons as the origin of this additional contribution to thermal conductivity. Moreover, this low temperature contribution prevails a definite conclusion on the validity or violation of the Wiedemann-Franz law at the field-induced quantum critical point.

Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2.

We investigate how the order parameter of a continuous phase transition can be protected from view by symmetry in a magnetic crystal. The symmetry in question forbids atomic displacements and formation of magnetic dipoles, rendering the order parameter invisible in standard x-ray and magnetic neutron Bragg diffraction. Analysis of the allowed magnetic space-groups reveals exact properties of the hidden order parameter. We demonstrate that Bragg spots forbidden by the chemical structure can unveil magnetic hidden order. The method is applied to URu2Si2, which has been thoroughly investigated in the past few decades using all manner of experimental techniques. Starting from the established chemical structure of URu2Si2, we have performed a critical analysis of available data for magnetic neutron Bragg diffraction.

Magnetic structure of phase II in U(Ru(0.96)Rh(0.04))2Si2 determined by neutron diffraction under pulsed high magnetic fields.

We report neutron diffraction measurements on U(Ru(0.96)Rh(0.04))(2)Si(2) single crystal under pulsed high magnetic fields up to 30 T applied along the tetragonal c axis. The high-field experiments revealed that the field-induced phase II above 26 T corresponds to a commensurate up-up-down ferrimagnetic structure characterized by the wave vector q=(2/3,0,0) with the magnetic moments parallel to the c axis, which naturally explains the one-third magnetization plateau and the substantially changed Fermi surface in phase II. This a-axis modulated magnetic structure indicates that the phase II near the hidden order phase is closely related to the characteristic incommensurate magnetic fluctuations at Q(1)=(0.6,0,0) in the pure system URu(2)Si(2), in contrast to the pressure-induced antiferromagnetic order at Q(0)=(1,0,0).

Metamagnetic transition in UCoAl probed by thermoelectric measurements.

We report field and temperature dependent measurements of the thermoelectric power (TEP) and the Nernst effect in the itinerant metamagnet UCoAl. The magnetic field is applied along the easy magnetization c axis in the hexagonal crystal structure. The metamagnetic transition from the paramagnetic phase at zero field to the field induced ferromagnetic state is of first order at low temperatures and becomes a broad crossover above the critical temperature T(M)(⋆)∼11 K. The field dependence of the TEP reveals that the effective mass of the hole carriers changes significantly at the metamagnetic transition. The TEP experiment reflects the existence of different carrier types in good agreement with band structure calculations and previous Hall effect experiments. According to the temperature dependence of the TEP, no Fermi liquid behavior appears in the paramagnetic state down to 150 mK, but is achieved only in the field induced ferromagnetic state.

Verification of the Wiedemann-Franz law in YbRh2Si2 at a quantum critical point.

The thermal conductivity measurements are performed on the heavy-fermion compound YbRh(2)Si(2) down to 0.04 K and under magnetic fields through a quantum critical point (QCP) at B(c)=0.66 T∥c axis. In the limit as T→0, we find that the Wiedemann-Franz law is satisfied within experimental error at the QCP despite the destruction of the standard signature of Fermi liquid. Our results place strong constraints on models that attempt to describe the nature of the unconventional quantum criticality of YbRh(2)Si(2).

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 ß-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 ß-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 ß-YbAlB4, but drastically reduced in YbRh2Si2. The ratio of thermopower to specific heat remains constant for ß-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.

Hidden order in URu2Si2 unveiled.

We report on measurements, by polarized neutron elastic scattering, of the magnetization distribution induced in a single crystal of URu2Si2 under a magnetic field applied along the tetragonal c axis. A subtle change in this distribution, revealed by maximum entropy analysis of the data, is found when the temperature is decreased to the range of the hidden order. An analysis in terms of U(4+) ionic states reveals that this change is a fingerprint of a freezing of rank 5 multipoles, i.e., dotriacontapoles.

Field evolution of coexisting superconducting and magnetic orders in CeCoIn5.

We present nuclear magnetic resonance (NMR) measurements on the three distinct In sites of CeCoIn5 with a magnetic field applied in the [100] direction. We identify the microscopic nature of the long range magnetic order (LRO) stabilized at low temperatures in fields above 10.2 T while still in the superconducting (SC) state. We infer that the ordered moment is oriented along the c axis and map its field evolution. The study of the field dependence of the NMR shift for the different In sites indicates that the LRO likely coexists with a modulated SC phase, possibly that predicted by Fulde, Ferrell, Larkin, and Ovchinnikov. Furthermore, we discern a field region dominated by strong spin fluctuations where static LRO is absent and propose a revised phase diagram.

Similarity of the Fermi surface in the hidden order state and in the antiferromagnetic state of URu2Si2.

Shubnikov-de Haas measurements of high quality URu2Si2 single crystals reveal two previously unobserved Fermi surface branches in the so-called hidden order phase. Therefore, about 55% of the enhanced mass is now detected. Under pressure in the antiferromagnetic state, the Shubnikov-de Haas frequencies for magnetic fields applied along the crystalline c axis show little change compared with the zero pressure data. This implies a similar Fermi surface in both the hidden order and antiferromagnetic states, which strongly suggests that the lattice doubling in the antiferromagnetic phase due to the ordering vector Q(AF)=(001) already occurs in the hidden order. These measurements provide a good test for existing or future theories of the hidden order parameter.

Tricritical point and wing structure in the itinerant ferromagnet UGe2.

Precise resistivity measurements on the ferromagnetic superconductor UGe2 under pressure p and magnetic field H reveal a previously unobserved change of the anomaly at the Curie temperature. Therefore, the tricritical point (TCP) where the paramagnetic-to-ferromagnetic transition changes from a second order to a first order transition is located in the p-T phase diagram. Moreover, the evolution of the TCP can be followed under the magnetic field in the same way. It is the first report of the boundary of the first order plane which appears in the p-T-H phase diagram of weak itinerant ferromagnets. This line of critical points starts from the TCP and will terminate at a quantum critical point. These measurements provide the first estimation of the location of the quantum critical point in the p-H plane and will inspire similar studies of the other weak itinerant ferromagnets.

Field re-entrant hidden-order phase under pressure in URu2Si2.

We succeeded in growing high quality single crystals of URu(2)Si(2) and performed thermal expansion measurements under pressure. Applying a magnetic field along the [001] direction in the tetragonal structure, the so-called hidden-order phase reappears after the suppression of the antiferromagnetic phase above the critical pressure P(x). We determined the pressure-temperature-field phase diagram for the paramagnetic, hidden-order and antiferromagnetic states for the [Formula: see text] direction. We also present the temperature dependence of the upper critical field H(c2) for [Formula: see text] and [100] determined by the AC specific heat measurements, corresponding to the bulk superconductivity in a high quality single crystal.

Excitation spectrum of PrOs(4)Sb(12) under a magnetic field.

The evolution of the magnetic excitation spectrum of the heavy fermion superconductor PrOs(4)Sb(12) was studied by inelastic neutron scattering on crossing the critical field H(c2) for superconductivity at low temperature. The peak positions in energy and the peak intensities of the modes of the triplet split by magnetic field confirm the known crystal field parameters for PrOs(4)Sb(12) in T(h) symmetry. A selective broadening of the lineshape occurs on increasing the magnetic field: the linewidth of the upper mode of the triplet increases while the one of the middle mode does not.

Magnetic-field dependence of the YbRh2Si2 Fermi surface.

Magnetic-field-induced changes of the Fermi surface play a central role in theories of the exotic quantum criticality of YbRh2Si2. We have carried out de Haas-van Alphen measurements in the magnetic-field range 8 T < or = H < or = 16 T, and directly observe field dependence of the extremal Fermi surface areas. Our data support the theory that a low-field "large" Fermi surface, including the Yb 4f quasihole, is increasingly spin split until a majority-spin branch undergoes a Lifshitz transition and disappears at H0 approximately 10 T, without requiring 4f localization at H0.

Multigap superconductivity in the heavy-Fermion system CeCoIn5.

New thermal conductivity experiments on the heavy-fermion superconductor CeCoIn5 down to 10 mK rule out the suggested existence of unpaired electrons. Moreover, they reveal strong multigap effects with a remarkably low "critical" field Hc2S for the small gap band, showing that the complexity of heavy-fermion band structure has a direct impact on their response under magnetic field.