Special Issue on Strongly Correlated Electron Systems

I present a brief review of the Landau Fermi-liquid theory of the normal and superfluid phases of liquid ${}_{{}}^{3}\text{He}$, with emphasis on its philosophy, its successes and its limitations.

Understanding the emergence and subsequent behavior of heavy electrons in Kondo lattice materials is one of the grand challenges in condensed matter physics. From this perspective we review the progress that has been made during the past decade and suggest some directions for future research. Our focus will be on developing a new microscopic framework that incorporates the basic concepts that emerge from a phenomenological description of the key experimental findings.

I describe early work on strongly correlated electron systems (SCES) from the perspective of a theoretical physicist who, while a participant in their reductionist top-down beginnings, is now part of the paradigm change to a bottom-up 'emergent' approach with its focus on using phenomenology to find the organizing principles responsible for their emergent behavior disclosed by experiment—and only then constructing microscopic models that incorporate these. After considering the organizing principles responsible for the emergence of plasmons, quasiparticles, and conventional superconductivity in SCES, I consider their application to three of SCES's sister systems, the helium liquids, nuclei, and the nuclear matter found in neutron stars. I note some recent applications of the random phase approximation and examine briefly the role that paradigm change is playing in two central problems in our field: understanding the emergence and subsequent behavior of heavy electrons in Kondo lattice materials; and finding the mechanism for the unconventional superconductivity found in heavy electron, organic, cuprate, and iron-based materials.

This article overviews the development of heavy-fermion superconductivity, notably in such rare-earth-based intermetallic compounds which behave as Kondo-lattice systems. Heavy-fermion superconductivity is of unconventional nature in the sense that it is not mediated by electron-phonon coupling. Rather, in most cases the attractive interaction between charge carriers is apparently magnetic in origin. Fluctuations associated with an antiferromagnetic (AF) quantum critical point (QCP) play a major role. The first heavy-fermion superconductor CeCu₂Si₂ turned out to be the prototype of a larger group of materials for which the underlying, often pressure-induced, AF QCP is likely to be of a three-dimensional (3D) spin-density-wave (SDW) variety. For UBe₁₃, the second heavy-fermion superconductor, a magnetic-field-induced 3D SDW QCP inside the superconducting phase can be conjectured. Such a 'conventional', itinerant QCP can be well understood within Landau's paradigm of order-parameter fluctuations. In contrast, the low-temperature normal-state properties of a few heavy-fermion superconductors are at odds with the Landau framework. They are characterized by an 'unconventional', local QCP which may be considered a zero-temperature 4 f-orbital selective Mott transition. Here, as concluded for YbRh₂Si₂, the breakdown of the Kondo effect concurring with the AF instability gives rise to an abrupt change of the Fermi surface. Very recently, superconductivity was discovered for this compound at ultra-low temperatures. Therefore, YbRh₂Si₂ along with CeRhIn₅ under pressure provide a natural link between the large group of about fifty low-temperature heavy-fermion superconductors and other families of unconventional superconductors with substantially higher T_c, e.g. the doped Mott insulators of the perovskite-type cuprates and the organic charge-transfer salts.

Heavy fermions have served as prototype examples of strongly-correlated electron systems. The occurrence of unconventional superconductivity in close proximity to the electronic instabilities associated with various degrees of freedom points to an intricate relationship between superconductivity and other electronic states, which is unique but also shares some common features with high temperature superconductivity. The magnetic order in heavy fermion compounds can be continuously suppressed by tuning external parameters to a quantum critical point, and the role of quantum criticality in determining the properties of heavy fermion systems is an important unresolved issue. Here we review the recent progress of studies on Ce based heavy fermion superconductors, with an emphasis on the superconductivity emerging on the edge of magnetic and charge instabilities as well as the quantum phase transitions which occur by tuning different parameters, such as pressure, magnetic field and doping. We discuss systems where multiple quantum critical points occur and whether they can be classified in a unified manner, in particular in terms of the evolution of the Fermi surface topology.

The term frustration refers to lattice systems whose ground state cannot simultaneously satisfy all the interactions. Frustration is an important property of correlated electron systems, which stems from the sign of loop products (similar to Wilson products) of interactions on a lattice. It was early recognized that geometric frustration can produce rather exotic physical behaviors, such as macroscopic ground state degeneracy and helimagnetism. The interest in frustrated systems was renewed two decades later in the context of spin glasses and the emergence of magnetic superstructures. In particular, Phil Anderson's proposal of a quantum spin liquid ground state for a two-dimensional lattice S  =  1/2 Heisenberg magnet generated a very active line of research that still continues. As a result of these early discoveries and conjectures, the study of frustrated models and materials exploded over the last two decades. Besides the large efforts triggered by the search of quantum spin liquids, it was also recognized that frustration plays a crucial role in a vast spectrum of physical phenomena arising from correlated electron materials. Here we review some of these phenomena with particular emphasis on the stabilization of chiral liquids and non-coplanar magnetic orderings. In particular, we focus on the ubiquitous interplay between magnetic and charge degrees of freedom in frustrated correlated electron systems and on the role of anisotropy. We demonstrate that these basic ingredients lead to exotic phenomena, such as, charge effects in Mott insulators, the stabilization of single magnetic vortices, as well as vortex and skyrmion crystals, and the emergence of different types of chiral liquids. In particular, these orderings appear more naturally in itinerant magnets with the potential of inducing a very large anomalous Hall effect.

The development of sensible microscopic models is essential to elucidate the normal-state and superconducting properties of the iron-based superconductors. Because these materials are mostly metallic, a good starting point is an effective low-energy model that captures the electronic states near the Fermi level and their interactions. However, in contrast to cuprates, iron-based high-T_c compounds are multi-orbital systems with Hubbard and Hund interactions, resulting in a rather involved 10-orbital lattice model. Here we review different minimal models that have been proposed to unveil the universal features of these systems. We first review minimal models defined solely in the orbital basis, which focus on a particular subspace of orbitals, or solely in the band basis, which rely only on the geometry of the Fermi surface. The former, while providing important qualitative insight into the role of the orbital degrees of freedom, do not distinguish between high-energy and low-energy sectors and, for this reason, generally do not go beyond mean-field. The latter allow one to go beyond mean-field and investigate the interplay between superconducting and magnetic orders as well as Ising-nematic order. However, they cannot capture orbital-dependent features like spontaneous orbital order. We then review recent proposals for a minimal model that operates in the band basis but fully incorporates the orbital composition and symmetries of the low-energy excitations. We discuss the results of the renormalization group study of such a model, particularly of the interplay between superconductivity, magnetism, and spontaneous orbital order, and compare theoretical predictions with experiments on iron pnictides and chalcogenides. We also discuss the impact of the glide-plane symmetry on the low-energy models, highlighting the key role played by the spin–orbit coupling.

Chiral superconductivity is a striking quantum phenomenon in which an unconventional superconductor spontaneously develops an angular momentum and lowers its free energy by eliminating nodes in the gap. It is a topologically non-trivial state and, as such, exhibits distinctive topological modes at surfaces and defects. In this paper we discuss the current theory and experimental results on chiral superconductors, focusing on two of the best-studied systems, Sr₂RuO₄, which is thought to be a chiral triplet p-wave superconductor, and UPt₃, which has two low-temperature superconducting phases (in zero magnetic field), the lower of which is believed to be chiral triplet f-wave. Other systems that may exhibit chiral superconductivity are also discussed. Key signatures of chiral superconductivity are surface currents and chiral Majorana modes, Majorana states in vortex cores, and the possibility of half-flux quantum vortices in the case of triplet pairing. Experimental evidence for chiral superconductivity from μSR, NMR, strain, polar Kerr effect and Josephson tunneling experiments are discussed.

The emergence of novel quantum ground states in correlated electron systems with strong spin–orbit coupling has been a recent subject of intensive studies. While it has been realized that spin–orbit coupling can provide non-trivial band topology in weakly interacting electron systems, as in topological insulators and semi-metals, the role of electron–electron interaction in strongly spin–orbit coupled systems has not been fully understood. The availability of new materials with significant electron correlation and strong spin–orbit coupling now makes such investigations possible. Many of these materials contain 5d or 4d transition metal elements; the prominent examples are iridium oxides or iridates. In this review, we succinctly discuss recent theoretical and experimental progress on this subject. After providing a brief overview, we focus on pyrochlore iridates and three-dimensional honeycomb iridates. In pyrochlore iridates, we discuss the quantum criticality of the bulk and surface states, and the relevance of the surface/boundary states in a number of topological and magnetic ground states, both in the bulk and thin film configurations. Experimental signatures of these boundary and bulk states are discussed. Domain wall formation and strongly-direction-dependent magneto-transport are also discussed. In regard to the three-dimensional honeycomb iridates, we consider possible quantum spin liquid phases and unusual magnetic orders in theoretical models with strongly bond-dependent interactions. These theoretical ideas and results are discussed in light of recent resonant x-ray scattering experiments on three-dimensional honeycomb iridates. We also contrast these results with the situation in two-dimensional honeycomb iridates. We conclude with the outlook on other related systems.

Kondo systems ranging from the single Kondo impurity to heavy fermion materials present us with a plethora of unconventional properties whose theoretical understanding is still one of the major open problems in condensed matter physics. Over the last few years, groundbreaking scanning tunneling spectroscopy (STS) experiments have provided unprecedented new insight into the electronic structure of Kondo systems. Interpreting the results of these experiments—the differential conductance and the quasi-particle interference spectrum—however, has been complicated by the fact that electrons tunneling from the STS tip into the system can tunnel either into the heavy magnetic moment or the light conduction band states. In this article, we briefly review the theoretical progress made in understanding how quantum interference between these two tunneling paths affects the experimental STS results. We show how this theoretical insight has allowed us to interpret the results of STS experiments on a series of heavy fermion materials providing detailed knowledge of their complex electronic structure. It is this knowledge that is a conditio sine qua non for developing a deeper understanding of the fascinating properties exhibited by heavy fermion materials, ranging from unconventional superconductivity to non-Fermi-liquid behavior in the vicinity of quantum critical points.

Since the announcement in 2011 of the Materials Genome Initiative by the Obama administration, much attention has been given to the subject of materials design to accelerate the discovery of new materials that could have technological implications. Although having its biggest impact for more applied materials like batteries, there is increasing interest in applying these ideas to predict new superconductors. This is obviously a challenge, given that superconductivity is a many body phenomenon, with whole classes of known superconductors lacking a quantitative theory. Given this caveat, various efforts to formulate materials design principles for superconductors are reviewed here, with a focus on surveying the periodic table in an attempt to identify cuprate analogues.

We review the recent developments in the field of cuprate superconductors with special focus on the recently observed charge order in the underdoped compounds. We introduce new theoretical developments following the study of the antiferromagnetic quantum critical point in two dimensions, in which preemptive orders in both charge and superconducting (SC) sectors emerge, that are in turn related by an SU(2) symmetry. We consider the implications of this proliferation of orders in the underdoped region, and provide a study of the type of fluctuations which characterize the SU(2) symmetry. We identify an intermediate energy scale where the SC fluctuations are dominant and argue that they are unstable towards the formation of a resonant excitonic state at the pseudogap temperature T ^*. We discuss the implications of this scenario for a few key experiments.

We review the theory of mixed-valent metals and make comparison with experiments. A single-impurity description of the mixed-valent state is discussed alongside the description of the nearly-integer valent or Kondo limit. The degeneracy N of the f-shell plays an important role in the description of the low-temperature Fermi-liquid state. In particular, for large N, there is a rapid cross-over between the mixed-valent and the Kondo limit when the number of f electrons is changed. We discuss the limitations on the application of the single-impurity description to concentrated compounds such as those caused by the saturation of the Kondo effect and those due to the presence of magnetic interactions between the impurities. This discussion is followed by a description of a periodic lattice of mixed-valent ions, including the role of the degeneracy N. The article concludes with a comparison of theory and experiment. Topics covered include the single-impurity Anderson model, Luttinger's theorem, the Friedel sum rule, the Schrieffer–Wolff transformation, the single-impurity Kondo model, Kondo screening, the Wilson ratio, local Fermi-liquids, Fermi-liquid sum rules, the Noziéres exhaustion principle, Doniach's diagram, the Anderson lattice model, the Slave-Boson method, etc.

The anomalous transport and thermodynamic properties in the quantum-critical region, in the cuprates, and in the quasi-two dimensional Fe-based superconductors and heavy-fermion compounds, have the same temperature dependences. This can occur only if, despite their vast microscopic differences, a common statistical mechanical model describes their phase transitions. The antiferromagnetic (AFM)-ic models for the latter two, just as the loop-current model for the cuprates, map to the dissipative XY model. The solution of this model in (2+1)D reveals that the critical fluctuations are determined by topological excitations, vortices and a variety of instantons, and not by renormalized spin-wave theories of the Landau–Ginzburg–Wilson type, adapted by Moriya, Hertz and others for quantum-criticality. The absorptive part of the fluctuations is a separable function of momentum $\mathbf{q}$, measured from the ordering vector, and of the frequency ω and the temperature T which scale as $\tanh (\omega /2T)$ at criticality. Direct measurements of the fluctuations by neutron scattering in the quasi-two-dimensional heavy fermion and Fe-based compounds, near their antiferromagnetic quantum critical point, are consistent with this form. Such fluctuations, together with the vertex coupling them to fermions, lead to a marginal fermi-liquid, with the imaginary part of the self-energy $\propto \text{max}(\omega,T)$ for all momenta, a resistivity $\propto T$, a $T\ln T$ contribution to the specific heat, and other singular fermi-liquid properties common to these diverse compounds, as well as to d-wave superconductivity. This is explicitly verified, in the cuprates, by analysis of the pairing and the normal self-energy directly extracted from the recent high resolution angle resolved photoemission measurements. This reveals, in agreement with the theory, that the frequency dependence of the attractive irreducible particle–particle vertex in the d-wave channel is the same as the irreducible particle-hole vertex in the full symmetry of the lattice.

The two-fluid model is a phenomenological description of the gradual change of the itinerant and local characters of f-electrons with temperature and other tuning parameters and has been quite successful in explaining many unusual and puzzling experimental observations in heavy electron materials. We review some of these results and discuss possible implications of the two-fluid model in understanding the microscopic origin of heavy electron physics.

This article attempts to review how band structure calculations can help to better understand the intriguing behavior of materials with strongly correlated electrons. Prominent examples are heavy-fermion systems whose highly anomalous low-temperature properties result from quantum correlations not captured by standard methods of electronic structure calculations. It is shown how the band approach can be modified to incorporate the typical many-body effects which characterize the low-energy excitations. Examples underlining the predictive power of this ansatz are discussed.

It has become increasingly feasible to use quantum Monte Carlo (QMC) methods to study correlated fermion systems for realistic Hamiltonians. We give a summary of these techniques targeted at researchers in the field of correlated electrons, focusing on the fundamentals, capabilities, and current status of this technique. The QMC methods often offer the highest accuracy solutions available for systems in the continuum, and, since they address the many-body problem directly, the simulations can be analyzed to obtain insight into the nature of correlated quantum behavior.

Electronic, magnetic, and structural phase inhomogeneities are ubiquitous in strongly correlated quantum materials. The characteristic length scales of the phase inhomogeneities can range from atomic to mesoscopic, depending on their microscopic origins as well as various sample dependent factors. Therefore, progress with the understanding of correlated phenomena critically depends on the experimental techniques suitable to provide appropriate spatial resolution. This requirement is difficult to meet for some of the most informative methods in condensed matter physics, including infrared and optical spectroscopy. Yet, recent developments in near-field optics and imaging enabled a detailed characterization of the electromagnetic response with a spatial resolution down to 10 nm. Thus it is now feasible to exploit at the nanoscale well-established capabilities of optical methods for characterization of electronic processes and lattice dynamics in diverse classes of correlated quantum systems. This review offers a concise description of the state-of-the-art near-field techniques applied to prototypical correlated quantum materials. We also discuss complementary microscopic and spectroscopic methods which reveal important mesoscopic dynamics of quantum materials at different energy scales.

We review some recent advances in the use of optical fields at terahertz frequencies to drive the lattice of complex materials. We will focus on the control of low energy collective properties of solids, which emerge on average when a high frequency vibration is driven and a new crystal structure induced. We first discuss the fundamentals of these lattice rearrangements, based on how anharmonic mode coupling transforms an oscillatory motion into a quasi-static deformation of the crystal structure. We then discuss experiments, in which selectively changing a bond angle turns an insulator into a metal, accompanied by changes in charge, orbital and magnetic order. We then address the case of light induced non-equilibrium superconductivity, a mysterious phenomenon observed in some cuprates and molecular materials when certain lattice vibrations are driven. Finally, we show that the dynamics of electronic and magnetic phase transitions in complex-oxide heterostructures follow distinctly new physical pathways in case of the resonant excitation of a substrate vibrational mode.

Nuclear magnetic resonance has emerged as a vital tool to explore the fundamental physics of Kondo lattice systems. Because nuclear spins experience two different hyperfine couplings to the itinerant conduction electrons and to the local f moments, the Knight shift can probe multiple types of spin correlations that are not accessible via other techniques. The Knight shift provides direct information about the onset of heavy electron coherence and the emergence of the heavy electron fluid.

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.

Magnetism and superconductivity compete or interact in complex and intricate ways. Here we review the special case where novel magnetic phenomena appear due to superconductivity, but do not exist without it. Such states have recently been identified in unconventional superconductors. They are different from the mere coexistence of magnetic order and superconductivity in conventional superconductors, or from competing magnetic and superconducting phases in many materials. We describe the recent progress in the study of such exotic magnetic phases, and articulate the many open questions in this field.

We review recent progress in point contact spectroscopy (PCS) to extract spectroscopic information out of correlated electron materials, with the emphasis on non-superconducting states. PCS has been used to detect bosonic excitations in normal metals, where signatures (e.g. phonons) are usually less than 1% of the measured conductance. In the superconducting state, point contact Andreev reflection (PCAR) has been widely used to study properties of the superconducting gap in various superconductors. It has been well-recognized that the corresponding conductance can be accurately fitted by the Blonder–Tinkham–Klapwijk (BTK) theory in which the AR occurring near the point contact junction is modeled by three parameters; the superconducting gap, the quasiparticle scattering rate, and a dimensionless parameter, Z, describing the strength of the potential barrier at the junction. AR can be as large as 100% of the background conductance, and only arises in the case of superconductors. In the last decade, there have been more and more experimental results suggesting that the point contact conductance could reveal new features associated with the unusual single electron dynamics in non-superconducting states, shedding a new light on exploring the nature of the competing phases in correlated materials. To correctly interpret these new features, it is crucial to re-examine the modeling of the point contact junctions, the formalism used to describe the single electron dynamics particularly in point contact spectroscopy, and the physical quantity that should be computed to understand the conductance. We will summarize the theories for point contact spectroscopy developed from different approaches and highlight these conceptual differences distinguishing point contact spectroscopy from tunneling-based probes. Moreover, we will show how the Schwinger–Kadanoff–Baym–Keldysh (SKBK) formalism together with the appropriate modeling of the nano-scale point contacts randomly distributed across the junction leads to the conclusion that the point contact conductance is proportional to the effective density of states, a physical quantity that can be computed if the electron self energy is known. The experimental data on iron based superconductors and heavy fermion compounds will be analyzed in this framework. These recent developments have extended the applicability of point contact spectroscopy to correlated materials, which will help us achieve a deeper understanding of the single electron dynamics in strongly correlated systems.

Low-dimensional electron systems fabricated from quantum matter have in recent years become available and are being explored with great intensity. This article gives an overview of the fundamental properties of such systems and summarizes the state of the field. We furthermore present and consider the concept of artificial atoms fabricated from quantum materials, anticipating remarkable scientific advances and possibly important applications of this new field of research. The surprising properties of these artificial atoms and of molecules or even of solids assembled from them are presented and discussed.

Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, $\sigma (\Omega,T)$. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) $\Omega /T$ scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron–electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, $1/\tau (\Omega,T)\propto {{ \Omega }^{2}}+4{{\pi}^{2}}{{T}^{2}}$, and show that a factor of $4{{\pi}^{2}}$ is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that $1/\tau (\Omega,T)$ must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, $1/\tau (\Omega,T)\propto {{ \Omega }^{2}}+b{{\pi}^{2}}{{T}^{2}}$, differs significantly from b  =  4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b  =  1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T  =  0: ${{\sigma}^{\prime}}(\Omega )\propto {{ \Omega }^{-1/3}}$ at low frequencies and ${{\sigma}^{\prime}}(\Omega )\propto {{ \Omega }^{-1}}$ at higher frequencies. We also discuss $\Omega /T$ scaling of the conductivity and show that ${{\sigma}^{\prime}}(\Omega,T)$ in the same model scales in a non-Fermi-liquid way, as ${{T}^{4/3}}{{ \Omega }^{-5/3}}$.

The realization of new classes of ground states in strongly correlated electron systems continues to be a major issue in condensed matter physics. Heavy fermion materials, whose electronic structure is essentially three-dimensional, are one of the most suitable systems for obtaining novel electronic states because of their intriguing properties associated with many-body effects. Recently, a state-of-the-art molecular beam epitaxy technique was developed to reduce the dimensionality of heavy electron systems by fabricating artificial superlattices that include heavy fermion compounds; this approach can produce a new type of electronic state in two-dimensional (2D) heavy fermion systems. In artificial superlattices of the antiferromagnetic heavy fermion compound CeIn₃ and the conventional metal LaIn₃, the magnetic order is suppressed by a reduction in the thickness of the CeIn₃ layers. In addition, the 2D confinement of heavy fermions leads to enhancement of the effective electron mass and deviation from the standard Fermi liquid electronic properties, which are both associated with the dimensional tuning of quantum criticality. In the superconducting superlattices of the heavy fermion superconductor CeCoIn₅ and nonmagnetic metal YbCoIn₅, signatures of superconductivity are observed even at the thickness of one unit-cell layer of CeCoIn₅. The most remarkable feature of this 2D heavy fermion superconductor is that the thickness reduction of the CeCoIn₅ layers changes the temperature and angular dependencies of the upper critical field significantly. This result is attributed to a substantial suppression of the Pauli pair-breaking effect through the local inversion symmetry breaking at the interfaces of CeCoIn₅ block layers. The importance of the inversion symmetry breaking in this system has also been supported by site-selective nuclear magnetic resonance spectroscopy, which can resolve spectroscopic information from each layer separately, even within the same CeCoIn₅ block layer. In addition, recent experiments involving CeCoIn₅/YbCoIn₅ superlattices have shown that the degree of the inversion symmetry breaking and, in turn, the Rashba splitting are controllable, offering the prospect of achieving even more fascinating superconducting states. Thus, these Kondo superlattices pave the way for the exploration of unconventional metallic and superconducting states.

Owing to a strong Coulomb repulsion, heavy electron superconductors mostly have anisotropic gap functions which have nodes for certain directions in the momentum space. Since the nodal structure is closely related to the pairing mechanism, its experimental determination is of primary importance. This article discusses the experimental methods of the gap determination by bulk heat capacity measurements in a rotating magnetic field. The basic idea is based on the fact that the quasiparticle density of states in the vortex state of nodal superconductors is field and direction dependent. We present our recent experimental results of the field-orientation dependence of the heat capacity in heavy fermion superconductors CeTIn₅ (T  =  Co, Ir), UPt₃, CeCu₂Si₂, and UBe₁₃ and discuss their gap structures.

Finding the exotic phenomena in strongly correlated electron systems (SCESs) and understanding the corresponding microphysics have long been the research frontiers of condensed matter physics. The remarkable examples for the intriguing phenomena discovered in past years include unconventional superconductivity, heavy Fermion behaviors, giant magneto-resistance and so on. A fascinating type of rare earth hexaboride RB₆ (R  =  Sm, Yb, Eu and Ce) belongs to a strongly correlated electron system (SCES), but shows unusual ambient-pressure and high-pressure behaviors beyond the phenomena mentioned above. Particularly, the recent discovery of the coexistence of an unusual metallic surface state and an insulating bulk state in SmB₆, known to be a Kondo insulator decades ago, by theoretical calculations and many experimental measurements creates new interest for the investigation of the RB₆. This significant progress encourages people to revisit the RB₆ with an attempt to establish a new physics that links the SCES and the unusual metallic surface state which is a common feature of a topological insulator (TI). It is well known that pressure has the capability of tuning the electronic structure and modifying the ground state of solids, or even inducing a quantum phase transition which is one of the kernel issues in studies of SCESs. In this brief review, we will describe the progress in high pressure studies on the RB₆ based on our knowledge and research interests, mainly focusing on the pressure-induced phenomena in YbB₆ and SmB₆, especially on the quantum phase transitions and their connections with the valence state of the rare earth ions. Moreover, some related high-pressure results obtained from CeB₆ and EuB₆ are also included. Finally, a summary is given in the conclusions and perspectives section.

The optical studies of high-temperature superconducting cuprates (HTSC) are reviewed. From the doping dependence of room temperature spectra, a dramatic change of the electronic state from a Mott (charge transfer) insulator to a Fermi liquid has been revealed. Additionally, the unusual 2D nature of the electronic state has been found. The temperature dependence of the optical spectra provided a rich source of information on the pseudogap, superconducting gap, Josephson plasmon, transverse Josephson plasma mode and precursory superconductivity. Among these issues, Josephson plasmons and transverse Josephson plasma mode were experimentally discovered by optical measurements, and thus are unique to HTSC. The effect of the spin/charge stripe order is also unique to HTSC, reflecting the conducting nature of the stripe order in this system. The pair-breaking due to the stripe order seems stronger in the out-of-plane direction than in the in-plane one.

In this article, we review the charge excitations of heavy fermion compounds probed by infrared spectroscopy. The article is not meant to be a comprehensive survey of experimental investigations. Rather it focuses on the dependence of charge excitations on the hybridization strength. In this context, the infrared properties of the Ce_mM_nIn_3m+2n family are discussed in detail since the hybridization strengths differ dramatically in different members despite their similar lattice structures. Investigations on some mixed valent compounds are also presented, aiming to elucidate the generic trend of the evolution. In particular, we address the scaling between hybridization energy gap ${{\Delta }_{\text{dir}}}$ and hybridization strength $\tilde{V}$($\propto \sqrt{W{{T}_{\text{K}}}}$) in a wide range of heavy fermion compounds, which demonstrates that the periodic Anderson model can generally and quantitatively describe the low-energy charge excitations.

An indispensable parameter to study strongly correlated electron systems is the magnetic field. Application of high magnetic fields allows the investigation, modification and control of different states of matter. Specifically for magnetic materials experimental tools applied in such fields are essential for understanding their fundamental properties. Here, we focus on selected high-field studies of frustrated magnetic materials that have been shown to host a broad range of fascinating new and exotic phases. We will give brief insights into the influence of geometrical frustration on the critical behavior of triangular-lattice antiferromagnets, the accurate determination of exchange constants in the high-field saturated state by use of electron spin resonance measurements, and the coupling of magnetic degrees of freedom to the lattice evidenced by ultrasound experiments. The latter technique as well allowed new, partially metastable phases in strong magnetic fields to be revealed.

Cerium hexaboride is a cubic f-electron heavy-fermion compound that displays a rich array of low-temperature magnetic ordering phenomena which have been the subject of investigation for more than 50 years. Its complex behaviour is the result of competing interactions, with both itinerant and local electrons playing important roles. Investigating this material has proven to be a substantial challenge, in particular because of the appearance of a 'magnetically hidden order' phase, which remained elusive to neutron-scattering investigations for many years. It was not until the development of modern x-ray scattering techniques that the long suspected multipolar origin of this phase was confirmed. Doping with non-magnetic lanthanum dilutes the magnetic cerium sublattice and reduces the f-electron count, bringing about substantial changes to the ground state with the emergence of new phases and quantum critical phenomena. To this day, Ce_1−xLa_xB₆ and its related compounds remain a subject of intense interest. Despite the substantial progress in understanding their behaviour, they continue to reveal new and unexplained physical phenomena. Here we present a review of the accumulated body of knowledge on this family of materials in order to provide a firm standpoint for future investigations.

A large swath of quantum critical and strongly correlated electron systems can be associated with the phenomena of preserved entropy and fragile magnetism. In this overview we present our thoughts and plans for the discovery and development of lanthanide and transition metal based, strongly correlated systems that are revealed by suppressed, fragile magnetism, quantum criticality, or grow out of preserved entropy. We will present and discuss current examples such as YbBiPt, YbAgGe, YbFe₂Zn₂₀, PrAg₂In, BaFe₂As₂, CaFe₂As₂, LaCrSb₃ and LaCrGe₃ as part of our motivation and to provide illustrative examples.

Strong electron correlations are at the heart of many physical phenomena of current interest to the condensed matter community. Here we present a survey of the mechanisms underlying such correlations in charge density wave (CDW) systems, including the current theoretical understanding and experimental evidence for CDW transitions. The focus is on emergent phenomena that result as CDWs interact with other charge or spin states, such as magnetism and superconductivity. In addition to reviewing the CDW mechanisms in 1D, 2D, and 3D systems, we pay particular attention to the prevalence of this state in two particular classes of compounds, the high temperature superconductors (cuprates) and the layered transition metal dichalcogenides. The possibilities for quantum criticality resulting from the competition between magnetic fluctuations and electronic instabilities (CDW, unconventional superconductivity) are also discussed.

We review the nature of strongly correlated electronic states in rare earth and actinide compounds, focusing on localized versus itinerant electronic states in CeRhIn₅, quantum critical phenomena in YbIr₂Zn₂₀, residual resistivity in CeCu₆, metamagnetism in heavy fermion compounds, and unconventional superconductivity in CeIrSi₃ without inversion symmetry in the crystal structure, emphasizing that sample quality is essentially important to clarify the characteristic features for the heavy fermion compounds.