The Topological Phase Transitions Related to Fermion Condensate

In this chapter, we consider so-called topological phase transitions, taking place in normal Fermi liquid. In other words, here we are dealing with different instabilities of normal Fermi liquids relative to several kinds of perturbations of initial quasiparticle spectrum \(\varepsilon (p)\) and occupation numbers \(n(p)\) associated with the emergence of a multi-connected Fermi surface. Depending on the parameters and analytical properties of the Landau interaction, such instabilities lead to several possible types of restructuring of initial Landau Fermi liquid ground state. This restructuring generates topologically distinct phases. One of them is the FC discussed above, another one belo…

Baryon Asymmetry Resulting from FCQPT in the Early Universe

This Chapter does not follow the main line of the book that is the theory of HF compounds but illustrates how the ideas of FC may be applicable to describe a very dissimilar system. Namely, here we consider a novel mechanism for explaining the matter-antimatter asymmetry of the universe. We assume that the universe starts from completely symmetric state and then, as it cools down, it undergoes a quantum phase transition, which in turn causes an asymmetry between matter and anti-matter. As we shall see the quantum phase transition is represented by FCQPT. The mechanism does not require the baryon number violating interactions or \({\textit{CP}}\) violation at a microscopic level. The state F…

Thermodynamic, dynamic and transport properties of quantum spin liquid in herbertsmithite from experimental and theoretical point of view

In our review we focus on the quantum spin liquid, defining the thermodynamic, transport and relaxation properties of geometrically frustrated magnets (insulators) represented by herbertsmithite $\rm ZnCu_{3}(OH)_6Cl_2$.

Highly Correlated Fermi Liquid in Heavy-Fermion Metals: Magnetic Properties

In this chapter we show how the FCQPT theory works, when describing the behavior of HF metals under the application of magnetic field. We show that a large body of experimental data regarding the thermodynamic, transport and relaxation properties collected in measurements on HF metals can be well explained. It is demonstrated that the experimental data exhibit the scaling behavior.

Theory of Heavy-Fermion Compounds : Theory of Strongly Correlated Fermi-Systems

Landau Fermi Liquid Theory and Beyond

In this chapter we consider the Landau theory of the Fermi liquid that has a long history and remarkable results in describing a numerous properties of the electron liquid in ordinary metals and Fermi liquids of the \(^3\)He type. The theory is based on the assumption that elementary excitations determine the physics at low temperatures, resembling that of weakly interacting Fermi gas. These excitations behave as quasiparticles with a certain effective mass. The effective mass \(M^*\) exhibits a simple universal behavior, for it is independent of the temperature, pressure, and magnetic field strength and is a parameter of the theory. Microscopically deriving the equation determining the eff…

Quantum Criticality of Heavy-Fermion Compounds

Chapter 17 is devoted to the quantum criticality of quantum spin liquids. In this chapter we continue to consider the nature of quantum criticality in HF compounds. The quantum criticality induced by the fermion condensation quantum phase transition extends over a wide range in the \(T-B\) phase diagram. As we shall see, the quantum criticality in all such different HF compounds, as high-\(T_c\) superconductors, HF metals, compounds with quantum spin liquids, quasicrystals, and 2D quantum liquids, is of the same nature. This challenging similarity between different HF compounds expresses universal physics that transcends the microscopic details of the compounds. This uniform behavior, induc…

Asymmetric Conductivity of Strongly Correlated Compounds

In this chapter, we show that the FC solutions for distribution function \(n_0(\mathbf{p})\) generate NFL behavior, and violate the particle-hole symmetry inherent in LFL. This, in turn, yields dramatic changes in transport properties of HF metals, particularly, the differential conductivity becomes asymmetric. As it is demonstrated in Sect. 3.1, Fermi quasiparticles can behave as Bose one. Such a state is viewed as possessing the supersymmetry (SUSY) that interchanges bosons and fermions eliminating the difference between them. In the case of asymmetrical conductivity it is the emerging SUSY that violates the time invariance symmetry. Thus, restoring one important symmetry, the FC state vi…

Violation of the Wiedemann-Franz Law in HF Metals

Experimental observations of the much-studied compounds CeCoIn\(_5\) and YbRh\(_2\)Si\(_2\) at vanishing temperatures carefully probe the nature of their magnetic-field-tuned QCPs. The violation of Wiedemann-Franz (WF) law, along with jumps revealed both in the residual resistivity \(\rho _0\) and the Hall resistivity \(R_H\), provide vital clues to the origin of their non-Fermi-liquid behavior. The empirical facts point unambiguously to association of the observed QCP with FC forming flat bands.

High Magnetic Fields Thermodynamics of Heavy Fermion Metals

In this chapter, we present the comprehensive theoretical analysis of thermodynamics of HF compounds at high magnetic fields. Such analysis permits to gain a deeper insight into the interplay of high magnetic field and temperature in suppressing and retrieving the Landau Fermi liquid state in these substances. Our analysis shows that although high magnetic fields and temperatures alter the properties of ordinary Landau quasiparticles, they survive, generating the experimentally observable anomalies in the thermodynamical quantities of HF compounds. We illustrate our theoretical findings by the example of the HF compound \(\mathrm{{YbRh_2Si_2}}\).

Metals with a Strongly Correlated Electron Liquid

In this chapter, we consider the main properties of strongly correlated Fermi systems, which are formed by the fermion condensate leading to the emergence of flat bands. Namely, we consider the residual entropy \(S_0\) related to the flat bands that leads to the violation of the quasiparticle—hole symmetry. The presence of \(S_0\) has a profound impact on the universality of second order phase transitions. In that case under the application of magnetic field the curve of the second order AF phase transitions passes into a curve of the first order ones at the tricritical point, thus leading to a violation of the critical universality of the fluctuation theory. We demonstrate that a jump in t…

Highly Correlated Fermi Liquid in Heavy-Fermion Metals: The Scaling Behavior

In this chapter we show how the FCQPT theory works. We do that on the base of experimentally relevant examples. Namely, as noted in the Introduction (Chap. 1), the challenge for the theories is to explain the scaling behavior of the normalized effective mass \(M^*_N(y)\) displayed in Fig. 1.3. The theories analyzing only the critical exponents characterizing \(M^*_N(y)\) at \(y\gg 1\) consider only a part of the problem. In this section we analyze and derive the scaling behavior of the normalized effective mass near QCP as reported in Fig. 1.3. We start with describing magnetic field dependence of the quasiparticle effective mass in Sect. 6.1. Quasiparticle damping and the temperature depen…

Quasi-classical Physics Within Quantum Criticality in HF Compounds

In this chapter, we explore how the fermion condensation paves the road for quasi-classical physics in HF compounds. This means simply that systems with FC admit partly the quasi-classical description of their thermodynamic and transport properties. This, in turn, simplifies a lot not only of their description but permits to gain more insights both in the puzzling NFL physics of HF compounds and of the physics of FC itself. The quasi-classical physics starts to be applicable near FCQPT, at which FC generates flat bands and quantum criticality, and makes the density of electron states in strongly correlated metals diverge. As we shall see, due to the formation of flat bands the strongly corr…

Magnetoresistance in the HF Metal at Zero Temperature

In this chapter we consider the paradoxical behavior of the residual resistivity \(\rho _0\) of HF metals in magnetic fields and under pressure. Our consideration is based on the idea of flattening of the single-particle spectrum \(\varepsilon (\mathbf{p})\) that profoundly effects on the specific heat \(C\), thermal expansion coefficient \(\alpha \) and magnetic susceptibility \(\chi \) in the normal state, the jump of \(C\) at the point of superconducting phase transition etc. We show that FC associated with flat bands contributes to the residual resistivity \(\rho _0\), while the application of the magnetic field or pressure to the system with a flat band removes the flat band and leads …

Fermion Condensation in Finite Systems

Here we consider another example of systems, in which fermion condensation takes place. These are what is called finite Fermi systems, i.e. systems with finite number of fermions, contrary to a solid, where the number of electrons is practically infinite. An example of a finite Fermi system is an atomic nucleus, having finite number of nucleons, protons and neutrons, which are fermions. Here we show that the fermion condensation manifests itself in finite Fermi systems as a forced merger of all, discreet for finite systems, single-particle levels, lying near the Fermi surface. On the first sight, this merger contradicts the standard Landau quasiparticle picture. Nevertheless, similar to inf…

Zero Temperature Magnetoresistance of the HF Metal: Enigma of $$\mathrm{Sr}_{3}\mathrm{Ru}_{2}\mathrm{O}_{7}$$

To understand the nature of field-tuned metamagnetic quantum criticality in the ruthenate \(\mathrm{Sr}_{3}\mathrm{Ru}_{2}\mathrm{O}_{7}\) is one of the significant challenges in the condensed matter physics. It is established experimentally that the entropy has a peak in the ordered phase. It is unexpectedly higher than that outside latter phase, while the magnetoresistivity varies abruptly near the ordered phase boundary. We demonstrate unexpected similarity between \(\mathrm{Sr}_{3}\mathrm{Ru}_{2}\mathrm{O}_{7}\) and HF metals expressing universal physics that transcends microscopic details. Our \(T-B\) phase diagram of \(\mathrm{Sr}_{3}\mathrm{Ru}_{2}\mathrm{O}_{7}\) explains main featu…

Fermi Liquid with Fermion Condensate

Here we discuss the general properties of FCQPT leading to the emergence of FC. We present a microscopic derivation of the main equations of FC, and show that Fermi systems with FC form an entirely new class of Fermi liquids with its own topological structure, protecting the FC state. We construct the phase diagram, and explore the order parameter of these systems. We show that the fermion condensate has a strong impact on the observable physical properties of systems, where it is realized, up to relatively high temperatures of a few tens kelvin. Two different scenarios of the quantum critical point (QCP), a zero-temperature instability of the Landau state, related to the divergence of the …

Appearance of Fermion-Condensation Quantum Phase Transition in Fermi Systems

As high-\(T_c\) superconductors are represented primarily by 2D layered structures, in Sect. 5.1 we discuss the superconducting state of a 2D liquid of heavy electrons, and within the framework of Gor’kov microscopic equations construct the Green functions of the FC state. On the other hand, our study can easily be generalized to the 3D case. To show that there is no fundamental difference between the 2D and 3D cases, we derive Green’s functions for the 3D case in Sect. 5.1.1. In Sect. 5.2, we consider the dispersion law and lineshape of single-particle excitations. Section 5.3 is devoted to the behavior of heavy-electron liquid with FC in magnetic field. In Sect. 5.4, we analyze conditions…

Quantum Criticality of Spin Liquids in Novel Insulators and Magnets

Strongly correlated Fermi systems are among the most intriguing and fundamental systems in physics, whose realization in some compounds is still under consideration. Quantum spin liquids are a promising new phases, where exotic quantum states of matter could be realized. Exotic quantum spin liquid (QSL) made of such hypothetic particles as fermionic spinons which carry spin \(1/2\) and no charge are considered in this chapter. Magnetic insulators with geometrical frustration produce important experimental facts shedding light on the nature of quantum spin liquid composed of spinons. We present a theory of the thermodynamic properties of quantum spin liquids, elucidating how their properties…