Magnetic Field as an Important Tool in Exploring the Strongly Correlated Fermi Systems and Their Particle–Hole and Time-Reversal Asymmetries

Abstract

In this review, we consider the impact of magnetic field on the properties of strongly correlated heavy-fermion compounds such as heavy-fermion metals and frustrated insulators with quantum spin liquid. Magnetic field B can be considered a universal tool, allowing the exploration of the physics controlling the remarkable properties of heavy-fermion compounds. These vivid properties are $T/B$ scaling, exhibited under the application of magnetic field B and at fixed temperature T, and the emergence of Landau Fermi liquid behavior under the application of magnetic field. We analyze the influence of quasiparticle–hole asymmetry on the properties of heavy-fermion (HF) compounds such as the universal scaling behavior of the thermopower $S/T$ exhibited under the application of magnetic field B. We show that universal scaling is demonstrated by different HF compounds such as $\beta$-${\mathrm{YbAlB}}_4$, ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, and strongly correlated layered cobalt oxide $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$. Analyzing ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, we show that the $T/B$ scaling behavior of $S/T$ is violated at the antiferromagnetic phase (AF) transition. The residual resistivity ${\rho }_0$ and the density of states $N_0$ experience jumps at the AF transition, causing two jumps in the thermopower and its sign reversal. Our consideration is based on the flattening of the single-particle spectrum that strongly affects ${\rho }_0$ and $N_0$ and leads to the violation of particle–hole symmetry. The particle–hole asymmetry generates the asymmetrical part $\mathsf{\Delta }{\sigma }_d\left(V\right)$ of tunneling differential conductivity ${\sigma }_d\left(V\right)$, $\mathsf{\Delta }{\sigma }_d\left(V\right)={\sigma }_d\left(V\right)-{\sigma }_d(-V)$, where V is the voltage bias. We demonstrate that in the presence of magnetic field, the quasiparticle–hole asymmetry vanishes, the LFL behavior is restored, and the asymmetry disappears. Our calculations of the mentioned properties of HF compounds, based on the fermion condensation theory, are in good agreement with the experiment and support our conclusion that the fermion condensation theory is capable of describing the properties of HF compounds, including those exhibited under the application of magnetic field.

1. Introduction

The Landau Fermi liquid (LFL) theory describes Fermi liquids such as ³He and electron liquids in ordinary metals. It is founded on the Landau paradigm where, at low temperatures, the properties of Fermi liquid are determined by Fermi quasiparticles. The number density x of quasiparticles coincides with that of particles in the Fermi liquid in question. Quasiparticles represent the elementary excitations of Fermi liquid, with the effective mass $M^{*}$ being a parameter of the theory, weakly dependent on temperature T, magnetic field B, etc. [1,2,3].

The discovery of strongly correlated Fermi systems represented by heavy-fermion (HF) compounds, such as heavy-fermion (HF) metals, high-$T_c$ superconductors, frustrated insulators with quantum spin liquid, and 2D ³He exhibiting non-Fermi liquid (NFL) behavior, has opened a new perspective in the area of modern condensed-matter physics [4,5,6,7,8,9,10,11,12]. Data collected on HF compounds demonstrate that the effective mass $M^{*}$ strongly depends on pressure P, T, x, B, etc., while $M^{*}$ itself can extend to very high values or even diverge [7,8]. This behavior is very unusual and cannot be described within the framework of the traditional Landau quasiparticles paradigm. The common opinion suggests that quantum criticality is induced by collective fluctuations, either magnetic or superconductive. Thus, HF compounds undergo a second-order phase transition at their quantum critical point (QCP). As a result of these superconducting or magnetic fluctuations, quasiparticles are suppressed, and the NFL behavior emerges, depending on the initial ground state, either magnetic or superconductive [4,5,7,8,10]. One expects that the NFL behavior is explained within the frameworks of the spin-density-wave scenario and the unconventional Kondo-breakdown scenario (see, e.g., [7,8,9]). Experimental facts reveal that these scenarios are not universal since HF compounds have very different microscopic and physical structures. Moreover, some HF compounds are not located at a quantum phase transition (QPT) with possible magnetic fluctuations, while others are represented by 2D ³He or by frustrated insulators with quantum spin liquid, which have nothing to do with fluctuations, the spin-density-wave scenario, or the Kondo effect.

The condensed-matter physics of the current millennium are vividly represented by the visiting card of the experimental discovery of flat bands (see, e.g., [11,12]) since HF compounds with flat bands are numerous [13]. Thus, one can expect the existence of a general physical mechanism facilitated by the presence of flat bands, exhibiting itself as the universal properties of HF compounds. As a result, HF compounds can demonstrate universal scaling behavior (see, e.g., [14,15,16,17]). Within the framework of the fermion condensation (FC) theory, such a mechanism is represented by the topological fermion condensation quantum phase transition (FCQPT) recreating quasiparticles, intimately related to the unlimited growth of $M^{*}$, producing NFL behavior, and generating flat bands. Further studies show that it is capable of delivering an adequate theoretical explanation of a vast majority of experimental results in different HF metals [14,15,16,17,18,19,20,21,22,23,24]. In contrast to the Landau paradigm, based on the assumption that $M^{*}$ is a constant, in the fermion condensation theory, the effective mass $M^{*}$ strongly depends on P, T, x, B, etc. As a consequence, to account for the numerous data, the extended quasiparticles paradigm should be employed. The main item here is that the well-defined quasiparticles define the thermodynamic, relaxation, and transport properties of HF compounds, making $M^{*}$ become a function of T, x, B, and P, etc. [15,23,25]. We note that the well-known Kadowaki–Woods relation [26] is conserved [15], and that obvious experimental facts can hardly be explained within the framework of the spin-density-wave scenario, the unconventional Kondo-breakdown scenario, etc. The FC theory has been successfully applied to describe the thermodynamic properties of different HF compounds. Thus, the outstanding puzzle of HF compounds originating from their universal behavior, which drastically differs from the behavior of ordinary metals and superconductors, is resolved [14,15,16,17,23,25], and the fundamental physics of HF compounds are controlled by the topological FCQPT (see, e.g., [15,16,17,20,27]). It is plausible to probe the other properties of HF compounds, which are not directly determined by the effective mass $M^{*}$ and cannot be explained within the framework of theories based on conventional quantum phase transitions (see, e.g., [15,20,24,28,29,30,31]). An important feature explained within the framework of the FC theory is the crossover from NFL behavior to LFL behavior under the application of magnetic field B, pressure P, etc. When the system in question transits under the application of magnetic field B from its NFL state to the LFL state, the impact of FC on the thermodynamic and transport properties is strongly changed. For example, the quasiparticle–hole asymmetry generating the asymmetrical part $\mathsf{\Delta }{\sigma }_d\left(V\right)$ of tunneling differential conductivity ${\sigma }_d\left(V\right)$ vanishes, where $\mathsf{\Delta }{\sigma }_d\left(V\right)={\sigma }_d\left(V\right)-{\sigma }_d(-V)$, and where V is the voltage bias [15]. Due to the same reason, the residual resistivity ${\rho }_0\left(B\right)$ under the application of magnetic field B is strongly reduced, and the behavior of the resistivity $\rho \left(T\right)$ changes from $\rho \left(T\right)\propto T$ to $\rho \left(T\right)\propto T^2$ [17,32]. Moreover, the dependence of the magnetic field on the Hall coefficient $R_H\left(B\right)$ provides information about the QCP, determining the properties of HF metals [31]. Experiments have shown that the Hall coefficient $R_H\left(B\right)$ in the antiferromagnetic HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ in magnetic fields B undergoes a jump in the zero-temperature limit under the application of magnetic field B, tuning the metal from the antiferromagnetic to the paramagnetic state at $B=B_{c0}$ [33]. The jump occurs when the magnetic field B reaches its critical value $B_{c0}$, at which the Néel temperature $T_N\left(B\right)$ of the antiferromagnetic transition vanishes, $T_N(B\to B_{c0})\to 0$. The jump is interpreted as a collapse of the large Fermi surface precisely at the QCP [33]. The FC theory successfully explains this behavior as a universal one represented by the transition from the NFL state with FC to the Landau Fermi liquid (LFL) state [17,31]. Moreover, the FC state is characterized by broken $\mathcal{T}$ and $\mathcal{C}$ symmetries occurring at the topological FCQPT in ${\mathrm{CeCoIn}}_5$ that connects two Fermi surfaces of different topological charges [15,16,17,19,20,30,31] (for detail, see Section 7.2). Recent measurements of the Hall coefficient on the prototypical superconductor ${\mathrm{CeCoIn}}_5$ allow one to interpret the observed quantum phase transition as a delocalization quantum phase transition without symmetry breaking in ${\mathrm{CeCoIn}}_5$. This transition is characterized by the delocalization of f-electrons in the transition that connects two Fermi surfaces of different volumes [28]. Measurements of $\mathsf{\Delta }{\sigma }_d\left(V\right)$ and ${\rho }_0\left(B\right)$ in the presence of the magnetic field B allow one to verify this observation.

In our review, based on important experimental facts and their explanation, we consider employing magnetic fields to reveal the physics of HF compounds. We focus on the crossover from the NFL behavior to the LFL behavior that takes place under the application of magnetic field B. This crossover is of crucial importance since at the LFL state, the quasiparticle particle–hole asymmetry generated by the FC state is eliminated. As a result, one can observe a change in the transport properties, such as the reduction of the residual resistivity ${\rho }_0\left(B\right)$, the vanishing of the asymmetrical part $\mathsf{\Delta }{\sigma }_d\left(V\right)$ of the differential tunneling conductivity ${\sigma }_d\left(V\right)$, and the reduction of the thermopower S. We also analyze the Hall coefficient $R_H\left(B\right)$ and explain its behavior in HF metals under the application of magnetic field. We demonstrate that this behavior is determined by the presence of flat bands formed by the topological FCQPT. The flat band in archetypal HF metal ${\mathrm{CeCoIn}}_5$ [34] forms its properties, while in the LFL state, the related violations of symmetries are eliminated. We show that the measured transport properties yield direct evidence of the apparent broken symmetries, such as the violation of the $\mathcal{T}$ and $\mathcal{C}$ symmetries occurring at the topological FCQPT in ${\mathrm{CeCoIn}}_5$. Analyzing the data collected in measurements of the change in the residual resistivity ${\rho }_0\left(B\right)$ on ${\mathrm{CeCoIn}}_5$, we demonstrate that the change in the Fermi surface of ${\mathrm{CeCoIn}}_5$ is determined by the topological FCQPT, connecting two Fermi surfaces of different topological charges. We show that it is the topological FCQPT that occurs in both the archetypal HF metals ${\mathrm{CeCoIn}}_5$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [17,31].

2. The Effective Mass M*

We start by considering an HF liquid at $T=0$ characterized by the effective mass $M^{*}$. Upon applying the well-known Landau equation, we relate $M^{*}$ with the bare electron mass M [2,35]

$$\frac{M^{*}}{M}=\frac{1}{1-N_0{F}^1\left(x\right)/3}.$$

(1)

Here, $N_0$ is the density of states of a free-electron gas, $x=p_F^3/3{\pi }^2$ is the number density, $p_F$ is the Fermi momentum, and $F^1\left(x\right)$ is the p-wave component of the Landau interaction amplitude F. At the critical point $x=x_c$, $F^1\left(x\right)$ reaches a certain value at which the denominator in Equation (1) vanishes. As a result, at $T=0$, the effective mass $M^{*}$ diverges, and the system is at the topological FCQPT (see, e.g., [15]). It is seen from Equation (1) that beyond the critical point $x_c$, the effective mass becomes negative. To prevent falling into the unstable and physically meaningless state with a negative effective mass, the system is to undergo the topological FCQPT occurring at the quantum critical point $x=x_c$ [15,20]. The asymmetrical phase behind the quantum critical point possesses the topological charge, forms a flat band, and is determined by [18,19,20]

$$\frac{\delta E\left[n\right(\mathbf{p}\left)\right]}{\delta n(\mathbf{p},T=0)}=\epsilon \left(\mathbf{p}\right)=\mu .$$

(2)

The solution of Equation (2) is delivered by the occupation numbers $n_0\left(\mathbf{p}\right)$ and takes place in the interval

$$0<n_0\left(\mathbf{p}\right)<1;p_i\le p\le p_f.$$

(3)

It is seen from Equation (2) that the flat band is located in the interval $p_i<p<p_f$. Here, E is the Landau functional of the ground-state energy [2], $\mu$ is a chemical potential, and $n\left(\mathbf{p}\right)$ is the occupation number of quasiparticles. The main result of such rearrangement is that instead of a Fermi step, we have $0<n_0\left(p\right)<1$ in a certain range of momenta $p_i\le p\le p_f$. Accordingly, the single-particle spectrum

$$\frac{\delta E}{\delta n\left(\mathbf{p}\right)}=\epsilon \left(\mathbf{p}\right)$$

(4)

in the momenta interval $(p_i-p_f)$ becomes flat, $\epsilon \left(\mathbf{p}\right)=\mu$, and this flat spectrum is known as the FC state [18]. It is seen from Equation (3) that the occupation numbers in the interval become $0<n_0\left(\mathbf{p}\right)<1$, and the FC state is characterized by the superconducting order parameter $\kappa \left(\mathbf{p}\right)=\sqrt{n_0\left(\mathbf{p}\right)(1-n_0\left(\mathbf{p}\right))}$ [15].

To obtain the equation defining the effective mass $M^{*}$, we employ the density functional theory of the superconducting state [36,37]. As a result, the ground-state energy E becomes the exact functional of the occupation numbers $n\left(\mathbf{p}\right)$ and the function of the number density x, $E=E\left[n\right(\mathbf{p}),x]$, while Equation (4) gives the exact single-particle spectrum $\epsilon \left(\mathbf{p}\right)$ [37]. Upon differentiating both sides of Equation (4) with respect to $\mathbf{p}$ and after some algebra and integration by parts, we obtain

$$\frac{\partial \epsilon \left(\mathbf{p}\right)}{\partial \mathbf{p}}=\frac{\mathbf{p}}{M}+\int F(\mathbf{p},{\mathbf{p}}_1)\frac{\partial n\left({\mathbf{p}}_1\right)}{\partial {\mathbf{p}}_1}\frac{d{\mathbf{p}}_1}{{\left(2\pi \right)}^3}.$$

(5)

Here, $F(\mathbf{p},{\mathbf{p}}_1)={\delta }^{2}E/\delta n\left(\mathbf{p}\right)\delta n\left({\mathbf{p}}_1\right)$ is the Landau amplitude. To calculate the derivative $\partial \epsilon \left(\mathbf{p}\right)/\partial \mathbf{p}$, we utilize the functional representation

$$\begin{array}{ccc}\hfill E\left[n\right]& =& \int \frac{p^2}{2M}n\left(\mathbf{p}\right)\frac{d\mathbf{p}}{{\left(2\pi \right)}^3}\hfill \\ & +& \frac{1}{2}\int F(\mathbf{p},{\mathbf{p}}_1)n\left(\mathbf{p}\right)n\left({\mathbf{p}}_1\right)\frac{d\mathbf{p}d{\mathbf{p}}_1}{{\left(2\pi \right)}^6}+...\hfill \end{array}$$

(6)

It follows from Equation (5) that $M^{*}$ is given by the well-known Landau equation

$$\frac{1}{M^{*}}=\frac{1}{M}+\int \frac{{\mathbf{p}}_F{\mathbf{p}}_{\mathbf{1}}}{p_F^3}F({\mathbf{p}}_{\mathbf{F}},{\mathbf{p}}_1)\frac{\partial n\left(p_1\right)}{\partial p_1}\frac{d{\mathbf{p}}_1}{{\left(2\pi \right)}^3}.$$

(7)

For simplicity, we have omitted the spin degrees of freedom. To calculate $M^{*}$ as a function of T, we construct the free energy $F=E-T{S}_e$, where the entropy $S_e$ is given by

$$S_e=-2\int [n\left(\mathbf{p}\right)ln\left(n\left(\mathbf{p}\right)\right)+(1-n\left(\mathbf{p}\right))ln(1-n\left(\mathbf{p}\right))]\frac{d\mathbf{p}}{{\left(2\pi \right)}^3},$$

(8)

that follows from general combinatorial reasoning [2]. Minimizing F with respect to $n\left(\mathbf{p}\right)$, we arrive at the Fermi–Dirac distribution

$$n(\mathbf{p},T)={\left\{1+exp\left[\frac{\left(\epsilon \right(\mathbf{p},T)-\mu )}{T}\right]\right\}}^{-1}.$$

(9)

Due to the employed procedure [37], we conclude that Equations (5) and (7) are exact and allow the calculation of the behavior of both $\partial \epsilon \left(\mathbf{p}\right)/\partial \mathbf{p}$ and $M^{*}$ in the vicinity of the topological FCQPT, where the well-defined quasiparticles determine the low-temperature physics, while $M^{*}$ becomes a function of T, B, and x. Effective mass $M^{*}$ diverges at $T\propto B\to 0$ and $x\to x_c$ [15,23,25]. This feature of $M^{*}$ forms the NFL behavior, and the crossover to the LFL behavior is observed in measurements on HF metals (see Figure 2a,b).

3. The Scaling of the Effective Mass

When analyzing HF compounds, we face the problem related to the anisotropy of the crystal lattice of solids, impurities, etc. This problem can be simplified, for we study the universal behavior of HF compounds based on the model of homogeneous HF liquid [15,16,17]. The model has a corresponding ability since we consider the universal scaling behavior exhibited by these materials at low temperatures a behavior related to the power-law divergences of quantities such as the effective mass, heat capacity, magnetization, etc. These divergences and the universal scaling of the effective mass or critical exponents that characterize them are set small compared to the Debye characteristic temperature and momentum transferred in energy and momentum on the order of the reciprocal lattice cell length $a^{-1}$. Therefore, quasiparticles are affected by the crystal lattice averaged over large distances compared to the length a. As a result, we can use the well-known jelly model to build a model of a homogeneous HF fluid [15,16,17].

In the FC theory, the QPT is interpreted as the topological FCQPT at which the quasiparticle effective mass $M^{*}$ diverges. In that case, quasiparticles with energy $\epsilon \sim \mu$ are well-defined excitations [15,16,17], while the FC state itself is protected by topological invariants [19,21,22]. In the vicinity of the FCQPT, in order to reveal the universal scaling behavior of $M^{*}$ observed in HF compounds, it is helpful to use “internal” scales for measuring quantities such as, e.g., $C/T$ and $M^{*}$, temperature T, etc. [15,16]. Maximum values ${(C/T)}_M\propto M_M^{*}$ in both $C/T$ and $M^{*}$, respectively, at temperature $T_M$ appear under the application of magnetic field B, and $T_M$ acquires higher values as magnetic field B is elevated. To obtain ${(C/T)}_N$, we use ${(C/T)}_M$ and $T_M$ as “internal” scales: ${(C/T)}_M$ is used to normalize $C/T$, and T is normalized by $T_M$ [15,16]. In the same way, we normalize $M^{*}$ to obtain the normalized effective mass $M_N^{*}=M^{*}/M_M^{*}$ as a function of the normalized temperature $T_N=T/T_M$. To study the scaling beahvior of $M^{*}(B,T)$, we use the model of homogeneous HF liquid, which permits the avoidance of complications associated with the crystalline anisotropy of solids [2,15,16]. We write the quasiparticle distribution function as $n_1\left(\mathbf{p}\right)=n(\mathbf{p},T)-n\left(\mathbf{p}\right)$, where $n\left(\mathbf{p}\right)$ is the step function, and Equation (7) then becomes [2,15,16]

$$\frac{1}{M^{*}\left(T\right)}=\frac{1}{M^{*}}+\int \frac{{\mathbf{p}}_F{\mathbf{p}}_{\mathbf{1}}}{p_F^3}F({\mathbf{p}}_{\mathbf{F}},{\mathbf{p}}_1)\frac{\partial n_1(p_1,T)}{\partial p_1}\frac{d{\mathbf{p}}_1}{{\left(2\pi \right)}^3}.$$

(10)

At the FCQPT, the effective mass $M^{*}$ diverges, and Equation (10) becomes homogeneous, determining $M^{*}$ as a function of temperature

$$M^{*}\left(T\right)\propto T^{-2/3},$$

(11)

while the system exhibits NFL behavior [15,16,25]. If the system is located before the topological FCQPT, $M^{*}$ is finite, and at low temperatures, the system demonstrates LFL behavior such that $M^{*}\left(T\right)\simeq M^{*}+a_1{T}^2$, where $a_1$ is a constant. The LFL regime occurs when the second term on the right-hand side of Equation (10) is small in comparison with the first term. Then, with rising temperatures, the system enters the transition regime: $M^{*}$ grows, reaching its maximum $M_M^{*}$ at $T=T_M$, and subsequently decreases. At $T\ge T_M$, the last “traces” of the LFL regime disappear, and the second term starts to dominate. Again, Equation (10) becomes homogeneous, and the NFL regime, manifesting itself in a decrease in $M^{*}$ as $T^{-2/3}$, is recreated. The application of magnetic field reproduces the LFL regime so that $M_M^{*}$ depends on B as

$$M_M^{*}\propto {(B-B_{c0})}^{-2/3},$$

(12)

while

$$T_M\propto {\mu }_B(B-B_{c0}),$$

(13)

where ${\mu }_B$ is the Bohr magneton [15,25]. Now, we consider the applied magnetic field B as a variable, while the temperature is fixed $T=T_f$. In that case, as seen from Equations (11) and (12), it is convenient to introduce variable $y=(B-B_{c0})/T$ and rewrite Equation (13) as [15]

$$y=T_N=\frac{T}{T_M}\propto \frac{T}{{\mu }_B(B-B_{c0})}.$$

(14)

Near the topological FCQPT, the normalized solution of Equation (10) $M_N^{*}\left(T_N\right)$ can be well approximated by a simple universal interpolating function [15,16]. The interpolation occurs between the LFL ($M^{*}\simeq M^{*}+a_1{T}^2$) and NFL ($M^{*}\propto T^{-2/3}$) regimes, describing the above crossover and representing the universal scaling of $M_N^{*}$ [15,16,25]

$$M_N^{*}\left(y\right)\approx c_0\frac{1+c_1{y}^2}{1+c_2{y}^{8/3}}.$$

(15)

Here, y is given by Equation (14), and $c_0=(1+c_2)/(1+c_1)$, $c_1$, and $c_2$ are fitting parameters. Magnetic field B enters Equation (10) only in the combination $B/T$, resulting in Equation (14). Thus, in the presence of a fixed magnetic field, the variable y becomes $y=T/T_M={(T/(B-B_{c0}))}_N\sim T/(B-B_{c0})$. Here, ${(T/(B-B_{c0}))}_N$ is the normalized value of $T/(B-B_{c0})$; it is calculated in the same way as it is in the case of $T_N$.

Thus, Equation (15) describes the universal scaling of $M_N^{*}$ as a function of T versus B. The curves $M_N^{*}$ at different magnetic fields B merge into a single curve in terms of the normalized variable $y=T/T_M$. The universal scaling of ${(C/T)}_N=M_N^{*}$ extracted from measurements of $C/T$ under the application of magnetic field of the archetypal HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [38] is shown in Figure 1. The solid theoretical curve coincides with that shown in Figure 2b. Equation (15) describes the scaling of $M_N^{*}(B,T)$ as a function of B versus T [2,15,16]. It is seen in Figure 1 that there are deviations between the theoretical curve and the normalized experimental data at $T_N\ge 7$, which can possibly come from our normalization of the experimental data. The deviation can also emerge due to the subtraction procedure of the contribution of the phonon to the electronic specific heat of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [38] at relatively high temperatures.

A schematic phase diagram of HF metal is shown in Figure 2a. Magnetic field B is taken as the control parameter. In fact, the control parameter can be pressure P or doping (the number density) x, etc., as well. At $B=B_{c0}$, FC occurs, leading to a strongly degenerated state, where $B_{c0}$ is a critical magnetic field such that, at $B>B_{c0}$, the system is driven towards its Landau Fermi liquid (LFL) regime. In our simple model, $B_{c0}$ is a parameter. The FC state is captured by the superconducting (SC), ferromagnetic (FM), antiferromagnetic (AFM), etc., states lifting the degeneracy [15,23]. Below, we consider the archetypal HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (see also Section 5). In this case, $B_{c0}\simeq 0.06$ T ($B\perp c$), and at $T=0$ and $B<B_{c0}$, the AFM state takes place at $T<T_{NL}\left(B\right)$ [39]. At elevated temperatures and a fixed magnetic field, the NFL regime occurs, while rising B again drives the system from the NFL region to the LFL region, as shown by the dashed-dotted horizontal arrow in Figure 2. Figure 2b demonstrates the universal scaling of the normalized effective mass $M_N^{*}=M^{*}/M_M^{*}$ versus normalized temperature $T_N=T/T_M={(T/B)}_N$, where $M_M^{*}$ is the maximum value that $M^{*}$ reaches at $T=T_M$. The $T^{-2/3}$ regime is marked as an NFL regime since the effective mass depends strongly on temperature. The temperature region $T\simeq T_M$ signifies the crossover between the LFL regime with almost constant effective mass and NFL behavior, given by $T^{-2/3}$ dependence. Thus, the region $T\sim T_M$ can be considered the crossover region taking place between the LFL and NFL regimes. From Equation (15), it follows that, in contrast to the Landau paradigm of quasiparticles, the effective mass $M^{*}(T,B)$ strongly depends on both temperature T and magnetic field B. We note that such a dependence $M^{*}(T,B)$ forms both the crossover region and the NFL region. It is also seen from Equation (15) that the scaling behavior of $M_N^{*}(T/B)$ near the topological FCQPT is formed by the absence of appropriate external physical scales to measure the effective mass and temperature. At fixed magnetic fields, the characteristic scales of the temperature and of the function $M^{*}(T,B)$ are defined by both $T_M$ and $M_M^{*}$, respectively. At fixed temperatures, the characteristic scales are $(B_M-B_{c0})$ and $M_M^{*}$. Here, $B_M$ is the magnetic field at which occurs the maximum $M_M^{*}$ of the effective mass. From Equations (12) and (13), it follows that at $T\simeq 0$ $T_M\to 0$, $M_M^{*}\to \infty$, and the width of the transition region shrinks to zero as $B\to B_{c0}$. In the same way, it is seen from Equations (11) and (14) that at a fixed magnetic field $(B-B_{c0})\simeq 0$, the maximum value $M_M^{*}\to \infty$ and the width of the transition region shrinks to zero as $T\to 0$.

It is seen in Figure 3 that the universal scaling behavior of the thermodynamic properties stems from the fact that HF compounds are located near the topological FCQPT, generating flat bands [15,16]. As an example, Figure 3a displays universal $B/T$ scaling for over more than three orders, demonstrated by $T^{2/3}\chi$ of the HF metal ${\mathrm{CeCu}}_{6-\mathrm{x}}{\mathrm{Au}}_{\mathrm{x}}$ and of the frustrated insulator herbertsmithite ${\mathrm{ZnCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2$. Here, $\chi$ is the magnetic susceptibility. Since the effective mass $M^{*}\left(T\right)$ diverges, $M^{*}\left(T\right)\propto T^{-2/3}$ (see Equation (11)), $T^{2/3}\chi \propto const$, as seen in Figure 3a. It is seen in Figure 3b that the normalized magnetic specific heat ${(C_{mag}/T)}_N$ exhibits scaling in a wide range of the variable ${(T/B)}_N$. The observed universal scaling exhibited by HF compounds represents a convincing example of the influence of magnetic field on the properties of the system, allowing the exploration of its universal properties. Data are extracted from Refs. [40,41,42]. The existence of such universal behavior, exhibited by various and very distinctive strongly correlated Fermi systems, supports the conclusion that HF compounds represent a new state of matter [15,16,17]. In contrast to the situation for an ordinary quantum phase transition, this scaling, induced by the topological FCQPT, occurs up to relatively high temperatures $T\sim 100$ K since the NFL behavior is defined by quasiparticles with $M_N^{*}$ given by Equation (15), rather than by fluctuations or Kondo lattice effects [15,16].

4. Flat Bands and Particle–Hole Asymmetry

Experimental facts show that the particle–hole asymmetry exists in HF compounds and is accompanied by Fermi surface transformation [43,44,45]. Such behavior is illuminated by the strong enhancement of the observed low-temperature value of the Seebeck coefficient [43,44,45]. The measurements on twisted bilayer graphene also confirm the strong particle–hole asymmetry that leads to the thermopower generation, so the Seebeck coefficient reaches high values of an order of magnitude larger than in common metals exhibiting LFL behavior. Both the observed asymmetry and the strong Seebeck coefficient enhancement were predicted about thirty years ago and determined by FC, forming flat bands [20]. As soon as the electronic system of the HF compound approaches the threshold of the topological FCQPT, the dispersion of an electronic quasiparticle disappears in some part of its spectrum, forming flat bands [15,16]. This dispersionless part is usually situated between momentum $p_i$ (standing for initial momentum) and $p_f$ (final momentum) (see Equation (2)). Clearly, in this dispersionless part of the spectrum, the effective mass $M^{*}$ diverges. Beyond the topological FCQPT, FC is formed so that the step quasiparticle distribution function $n_F(p,T\to 0)=\theta (p-p_F)$ does not minimize the Landau functional $E\left[n\right(\mathbf{p}\left)\right]$, and the quasiparticle distribution is determined by Equation (2) for the minimum of the functional. At $T=0$, Equation (2) determines the quasiparticle distribution function $n_0\left(\mathbf{p}\right)$, minimizing the ground-state energy $E\equiv E\left[n\right(\mathbf{p}\left)\right]$. Figure 4 reports a typical $n_0\left(\mathbf{p}\right)$, with $E_F$ being the Fermi energy. It is seen in Figure 4 that the particle–hole symmetry is violated by FC since $n_0\left(\mathbf{p}\right)$ does not evolve from the Fermi–Dirac distribution function, a step at $T=0$, shown by the blue line. In Figure 4, the arrow shows the hole states formed by the FC. As we will see in Section 5.2 and Section 7.2, as soon as the flat band is eliminated under the application of magnetic field, the asymmetry vanishes, and the thermopower is strongly diminished, while the residual resistivity ${\rho }_0$ strongly diminishes (see Section 7.1), and the asymmetrical part $\mathsf{\Delta }{\sigma }_d\left(V\right)$ of the tunneling conductivity disappears (see Section 7.2). These observations shed light on the role of the magnetic field B and strongly support the FC theory, explaining the emergence of the particle–hole asymmetry and the corresponding experimental facts collected on HF metals and graphene.

5. Thermopower under the Application of Magnetic Fields

The study of the thermopower $S/T$ in HF metals and graphene reveals a strong interaction driven particle–hole asymmetry [43,44,45]. In this section, we make use of data collected on the archetypal HF ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ in measurements of the thermopower [46,47] to illustrate the particle–hole asymmetry and its behavior in magnetic fields. Thermopower $S/T$ is a sensitive and helpful tool to analyze the quasiparticle excitations at the Fermi surface. Thus, we face an important problem related to revealing the scaling of the thermopower $S/T$, which permits the analysis of the nature of quasiparticle (electronic) excitations at the Fermi surface. In this direction, we will elucidate the role of quasiparticles and the nature of the topological FCQPT that forms the flat bands and both the particle–hole asymmetry and the behavior of the thermopower $S/T$ in different HF compounds. We show that the thermopower $S/T$ of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ exhibits scaling that coincides with that of other thermodynamic functions such as ${(C/T)}_N$. We demonstrate that $S/T$ of different HF compounds such as ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, $\beta$-${\mathrm{YbAlB}}_4$, and $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$ exhibit universal scaling, determined by the scaling behavior of the effective $M_N^{*}$ shown in Figure 1, Figure 2b and Figure 5. Using ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ as an example, we also demonstrate that the universal behavior of $S/T$ is violated at the AF phase transition, while the residual resistivity ${\rho }_0$ and the density of states $N_0$ experience downward jumps at the AF phase transition that are accompanied by the restoration of the particle–hole symmetry. The restoration of the particle–hole symmetry leads to the corresponding downward jumps ${\mathrm{Jump}}_{\mathrm{F}}$ of $S/T$ that strongly diminishes and changes its sign, as seen in Figure 6. This first jump, occurring at $T\le 0.03$ K, is shown in Figure 6 and is labeled ${\mathrm{Jump}}_{\mathrm{F}}$. It is accompanied by a change in sign of ${(S/T)}_N$, becoming positive [46,47]. This behavior occurs as the system undergoes the AF phase transition taking place at $T_{NL}$. This first jump ${\mathrm{Jump}}_{\mathrm{F}}$ reflects the elimination of both the hole state and the flat band in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (see Figure 4) [15,16,17].

5.1. Scaling

As we shall see, an analysis of the thermoelectric power $S/T$ delivers new knowledge of the nature of quantum phase transition defining the NFL behavior of the corresponding HF compounds. For example, one reasonably proposes that $S/T$ distinguishes between the two possible scenarios for HF metals, namely the spin-density-wave theory and the breakdown of the Kondo effect [48,49]. Indeed, $S/T$ is sensitive to the derivative of the density of electronic states $N_0$ and to the change in the relaxation time $\tau$ at $\mu$ [3,50]. It follows from the Boltzmann equation that $S/T$ can be written as [3,50,51,52,53]

$$\frac{S}{T}=-\frac{{\pi }^2{k}_B^2}{3e}{\left[\frac{\partial ln\sigma \left(\epsilon \right)}{\partial \epsilon }\right]}_{\epsilon =\mu },$$

(16)

where $k_B$ and e are, respectively, the Boltzmann constant and the electron’s charge, while $\sigma$ is the dc electric conductivity of the system:

$$\sigma \left(\epsilon \right)=2e^2\tau \left(\epsilon \right)\int \delta (\mu -\epsilon \left(\mathbf{p}\right))v\left(\mathbf{p}\right)v\left(\mathbf{p}\right)\frac{d\mathbf{p}}{{\left(2\pi \right)}^3},$$

(17)

$\mathbf{p}$ is the wave vector of the electron, and v is its velocity. It is seen from Equation (17) that $S/T$ depends on the derivative of the density of electronic states $N_0(\epsilon =\mu )$ and the change in the relaxation time at $\epsilon =\mu$. Within the framework of the Fermi liquid theory, the term in the brackets on the right-hand side of Equation (16) can be simplified. As a result, one obtains $S/T\propto N(\epsilon =\mu )\propto C/T\propto M^{*}$ at $T\to 0$ [50,51,52,53]. Taking into account that the charge and heat fluxes at low temperatures are transferred by quasiparticles, the $S/C$ ratio has the form

$$(S/C)\simeq (S/S_{ent})\simeq const.$$

(18)

Here, $S_{ent}$ is the entropy density of charge carriers [50,51,52,53]. It is seen from Equation (18) that $S/T\propto M^{*}$. Thus, within the FC theory, which proposes the universal scaling behavior of $M^{*}$ (see Section 3), one can obtain a reliable description of the thermopower and its universal scaling behavior [15,16,17,18,20,54,55].

Figure 5. Scaling of the susceptibility $\chi (B-B_{c0})$ as a function of scaled temperature $T/(B-B_{c0})$ with $B_{c0}=0.176$ T for various B values obtained in measurements on $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$ (see the legend [56]). The LFL region, crossover region, and NFL region are shown by the arrows. The solid curve represents our calculations based on Equation (10), and describes the universal scaling behavior ${(C/T)}_N=M_N^{*}\propto \chi (B-B_{c0})$ shown in Figure 2b.

Figure 5. Scaling of the susceptibility $\chi (B-B_{c0})$ as a function of scaled temperature $T/(B-B_{c0})$ with $B_{c0}=0.176$ T for various B values obtained in measurements on $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$ (see the legend [56]). The LFL region, crossover region, and NFL region are shown by the arrows. The solid curve represents our calculations based on Equation (10), and describes the universal scaling behavior ${(C/T)}_N=M_N^{*}\propto \chi (B-B_{c0})$ shown in Figure 2b.

Indeed, Figure 1 demonstrates the normalized specific heat ${(C/T)}_N$ as a function of $T_N$ measured on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. The behavior of ${(C/T)}_N$ coincides with that of the magnetic susceptibility $\chi \propto C/T\propto M^{*}$ obtained in measurements on $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$ [56] and shown in Figure 5. The solid curve in Figure 5 demonstrates results of the same calculations based on Equation (10), describing the universal scaling behavior ${(C/T)}_N(T/T_M)=M_N^{*}(T/T_M)\propto \chi {(B-B_{c0})}^{0.6}$ shown in Figure 1. As a result, we conclude that the solid curve drawn in both Figure 1 and Figure 5 exhibits the universal scaling intrinsic to HF compounds [15,16,17].

Figure 6. Thermopower isotherm $-S\left(B\right)/T$ for different temperatures given in the legend [46,47]. The label ${\mathrm{Jump}}_{\mathrm{F}}$ represents the first downward jumps in $-S\left(B\right)/T$ shown by the arrows. The solid line is a guide to the eye.

Figure 6. Thermopower isotherm $-S\left(B\right)/T$ for different temperatures given in the legend [46,47]. The label ${\mathrm{Jump}}_{\mathrm{F}}$ represents the first downward jumps in $-S\left(B\right)/T$ shown by the arrows. The solid line is a guide to the eye.

As seen in Figure 7, in the case of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ and at $T\ge T_{NL}$, the normalized isotherms ${(S\left(B\right)/T)}_N$ behave such as ${(C/T)}_N$ (see Figure 1). They demonstrate a broad maximum, which sharpens and shifts towards smaller fields upon cooling [46,47]. It is seen in Figure 6 that the above-mentioned behavior is violated as the system approaches the AF phase transition. Here, $T_{NL}\left(B\right)$ is the temperature of antiferromagnetic (AF) ordering, with $T_{NL}(B=0)=70$ mK, and $T_{NL}(B=B_{c0})=0$ at the critical field $B_{c0}=60$ mT, applied perpendicular to the magnetically hard c axis [57]. Thus, outside of the AF region, $S/T\propto C/T\propto \chi \propto M^{*}$ over a wide range of T and B since in the framework of FC theory, the particle–hole symmetry is violated [15,16,17]. It is worth noting that $S/T\propto M^{*}$ in a poorly ordered two-dimensional electron system in silicon and tends to diverge at a finite disorder-independent density [58].

To elucidate the scaling of the thermopower $S/T\propto C/T\propto M^{*}$, we normalize $S/T$ in the same way as in the normalization of $C/T$ (see Section 3 and Figure 1). Taking into account that $S/T\propto C/T$ [51,52,53,54], we conclude that ${(S/T)}_N={(C/T)}_N=M_N^{*}$, provided that the system in question is located away from possible phase transitions that eliminate the particle–hole asymmetry. This function ${(C/T)}_N=M_N^{*}$ is displayed in Figure 5. Figure 7a,b report ${(S/T)}_N$ as a function of the normalized magnetic field $B_N$ and $T_N$, respectively. In Figure 7a, the function ${(S/T)}_N$ is obtained by normalizing $(S/T)$ by its maximum, occurring at $B_M$, and the field B is scaled by $B_M$. As seen from Equation (15) and Figure 7a, the LFL regime sets in at $B_N>1$ since ${(S/T)}_N=M_N^{*}$, and $M_N^{*}\propto {(B-B_{c0})}^{-2/3}$ are T-independent, while at $B_N<1$, $M_M^{*}$ becomes T-dependent and exhibits NFL behavior with $M_N^{*}\propto T_N^{-2/3}$. As seen in Figure 7a,b, the theoretical curves representing the universal function $M_N^{*}$ (see Figure 1 and Figure 2b) are in good agreement with the corresponding data over the wide range of $B_N$ and $T_N$. Thus, ${(S/T)}_N={(C/T)}_N=M_N^{*}$ exhibits the universal scaling over a wide range of its scaled variables $B_N$ and $T_N$. Figure 7a also displays the scaling violation at the AF phase, when $B\le B_{c0}$. It is seen in Figure 6 and Figure 7a,b that the scaling is violated at $T\le T_{NL}$ by the two downward jumps. The second jump, shown in Figure 7a and marked as ${\mathrm{Jump}}_{\mathrm{S}}$, occurs at $T_{NL}>T>0.3$ K, while the first, taking place at $T\le 0.03$ K, is shown in Figure 6 as ${\mathrm{Jump}}_{\mathrm{F}}$. The latter is accompanied by a change in the sign of ${(S/T)}_N$, which now becomes positive [46,47]. Below, we shall see that the two jumps reflect the presence of a flat band at $\mu$ in the single-particle spectrum $\epsilon \left(\mathbf{p}\right)$ of HF in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [15,16,17]. As seen in Figure 7b, scaling is violated in the same way by the superconducting (SC) phase transition, occurring in $\beta$-${\mathrm{YbAlB}}_4$ at $T_c\simeq$ 80 mK [59].

It is worth noting that the universal scaling of ${(S/T)}_N$ is seen in the experimental data on the thermopower for $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$ [56]. The solid curve, representing our theory in Figure 8, is the same as that displaced in Figure 5, and describes ${(C/T)}_N$ extracted from measurements on the archetypal HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (see Figure 1 [15]). The universal scaling behavior, and the three regimes are seen in Figure 8. These regimes are in good agreement with those reporting the universal behavior displayed in Figure 2b, Figure 5 and Figure 7a,b.

5.2. Flat Bands and $S/T$ Jumps in the AF Phase Transition

As seen in Figure 4, the ground states of systems with flat bands are degenerate. As a result, the occupation numbers $n_0\left(\mathbf{p}\right)$ of single-particle states belonging to the flat band are given by a continuous function on the interval $[0,1]$. This behavior of the occupation numbers leads to the emergence of the particle–hole asymmetry, occurring in contrast to the LFL restriction to occupation numbers 0 and 1, resulting in the particle–hole symmetry [2]. Thus, the FC state generates the entropy excess

$$S_0=-\sum n_0\left(\mathbf{p}\right)ln{n}_0\left(\mathbf{p}\right)+(1-n_0\left(\mathbf{p}\right))ln(1-n_0\left(\mathbf{p}\right)).$$

(19)

The excess $S_0$ does not contribute to the specific heat $C\left(T\right)$. The existence of $S_0$ contradicts the Nernst theorem. To avoid the violation of the Nernst theorem, FC must be absorbed by a phase transition at $T\to 0$. For example, it can be the AF transition that becomes of the first order at some tricritical point occurring at $T=T_{\mathrm{tr}}$ (see Section 6) [15,16,17]. Such a first-order phase transition eliminates the flat portion in the single-particle spectrum. Therefore, at this first-order phase transition, both the high density of states generated by the flat band and the hole states, shown by the arrow in Figure 4, vanish discontinuously, while both the occupation numbers $n_0\left(\mathbf{p}\right)$ and the spectrum $\epsilon \left(\mathbf{p}\right)$ represent the LFL state, as shown by the arrows in Figure 7a. In the same way, the Fermi sphere undergoes an abrupt change, related to the first-order phase transition [15], in the interval from the momentum $p_i$ to $p_f$, so as to nullify both the swelling of the Fermi surface, the entropy excess $S_0$, defining the entropy $S_{ent}$ at low temperatures, and the particle–hole asymmetry. This is a vivid example of the influence of magnetic field, which drives the AF phase transition and allows the exploration of the properties of the system. As a result, the thermopower given by Equation (18) experiences ${\mathrm{Jump}}_{\mathrm{F}}$, as it is follows from Equation (19), for the entropy $S_0$ abruptly nullifies, while both the flat band and the particle–hole asymmetry vanish. We note that the jump is observed in the behavior of the low-T Hall coefficient of the HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [15,16,57,60].

It is seen in Figure 6 that at $T=0.03$ K, $S/T$ abruptly changes its sign (the first jump—Jump_F) because the hole states are eliminated. The positive sign of $S/T$ of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ without the hole states [52] corresponds to the positive thermopower of its nonmagnetic counterpart ${\mathrm{LuRh}}_2{\mathrm{Si}}_2$, which has no the $4f$ hole states at the chemical potential $\mu$ [46,47,61]. Contrarily, at $T_{NL}>T>T_{cr}$, the AF phase transition is of the second order, and the entropy is a continuous function at the border of the phase transition. Thus, during a second-order phase transition, both the occupation numbers and the spectrum remain the same and retain their FC-like shape, while the system with FC is destroyed, turning into HF liquid. This destruction generates the second jump ${\mathrm{Jump}}_{\mathrm{S}}$, shown in Figure 7. As a result, the FC state is destroyed and its contribution, ${\rho }_0^{FC}$, to the residual resistivity ${\rho }_0$ vanishes, resulting in a change in the scattering time $\tau (\epsilon =\mu )$. We note that in the presence of FC, the residual resistivity is represented by two terms ${\rho }_0={\rho }_0^{FC}+{\rho }_0^{imp}$ (see Section 7.1). Here, the residual resistivity ${\rho }_0^{FC}$ is formed by the flat band generated by the FC state, while the resistivity ${\rho }_0^{imp}$ is generated by impurities [16,32]. Therefore, the thermopower experiences the second jump ${\mathrm{Jump}}_{\mathrm{S}}$, as seen from Equations (16) and (17). The first downward jump ${\mathrm{Jump}}_{\mathrm{F}}$ under decreasing B, defined by elimination of both ${\rho }_0^{FC}$ and the hole states, is deeper than the second jump ${\mathrm{Jump}}_{\mathrm{S}}$ and leads to the change in the sign of $S/T$. This is consistent with the experimental observations, as seen in Figure 6 and Figure 7.

6. The Tricritical Point in the B − T Phase Diagram of YbRh₂Si₂

The Landau theory of the second-order phase transitions is valid near the tricritical point, occurring at $T\simeq T_{cr}$. In that case, the fluctuation theory generates only the logarithmic corrections to the values of the critical indices [62]. Upon using the Landau theory, we obtain that the Sommerfeld coefficient ${\gamma }_0=C/T$ takes the form ${\gamma }_0\propto {|t-1|}^{-\alpha }$, where $t=T/T_{NL}\left(B\right)$ with the exponent being $\alpha \simeq 0.5$ as the tricritical point is approached at fixed magnetic field B [17,62,63]. As seen in Figure 9, the value $\alpha =0.5$ gives a good description of the data collected in measurements of the specific heat on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [63]. Taking into account that the specific heat capacity increases during the transition from the symmetric AF phase to the asymmetric one [62], we obtain

$${\gamma }_N=\frac{{\gamma }_0\left(t\right)}{A_1}=1+\frac{B_1}{A_1\sqrt{|t-1|}}.$$

(20)

Here, $B_1=B_{\pm }$ are the proportionality factors which are different for the two sides of the phase transition. The parameters $A_1=A_{\pm }$, related to the corresponding specific heat ${(C/T)}_{\pm }$, are also different for the two sides, and “+” stands for $t>1$, while “−” stands for $t<1$.

Figure 9. The normalized Sommerfeld coefficient ${\gamma }_N={\gamma }_0/A_{\pm }$ as a function of the normalized temperature $t=T/T_{N0}$. The Sommerfeld coefficient ${\gamma }_0$ is given by Equation (20) and shown by the solid curves. The normalized Sommerfeld coefficient is extracted from the data obtained during measurements on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ at the AF phase transition [63] and is represented by geometric figures.

Figure 9. The normalized Sommerfeld coefficient ${\gamma }_N={\gamma }_0/A_{\pm }$ as a function of the normalized temperature $t=T/T_{N0}$. The Sommerfeld coefficient ${\gamma }_0$ is given by Equation (20) and shown by the solid curves. The normalized Sommerfeld coefficient is extracted from the data obtained during measurements on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ at the AF phase transition [63] and is represented by geometric figures.

The fit to the data for ${\gamma }_0=C\left(T\right)/T$ in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ at the AF phase transition in zero magnetic fields [63] is displayed in Figure 9. We show there the normalized Sommerfeld coefficient ${\gamma }_0/A_{+}$ as a function of the normalized temperature $t=T/T_{N0}$. It can be seen that the normalized Sommerfeld coefficient ${\gamma }_0/A_{+}$ extracted from $C/T$ measurements at ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [63] are well described over the entire temperature range around the antiferromagnetic phase transition by the formula (20) with $A_{+}=1$. Now, transform Equation (20) to the form

$${\gamma }_{norm}=\frac{{\gamma }_0\left(t\right)-A_1}{B_1}=\frac{1}{\sqrt{|t-1|}}.$$

(21)

It follows from Equation (21) that the ratios ${\gamma }_{norm}$ for $t<1$ and $t>1$ versus $|1-t|$ collapse into a single line on a double logarithmic plot. The ratios extracted from the data [63] are shown in Figure 10, the coefficients $A_1$ and $B_1$ are taken from fitting ${\gamma }_N$ shown in Figure 7. As seen in Figure 10, the ratio ${\gamma }_{norm}$ collapses into the single line.

Based on the fits to the data, given by Functions (20) and (21) (with the critical exponent $\alpha =1/2$) and shown in Figure 9 and Figure 10, we predict that the AF phase transition of the second order in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ goes into the first order under the action of a magnetic field [17]. As seen in Figure 9, the peak is sharp, while one expects that the anomalies in the specific heat related to the onset of magnetic order are to be broad [63]. Thus, the observed behavior demonstrates that the second-order phase transition is changed to the first-order phase transition at the tricritical point (see Figure 11).

Schematic $T-B$ Phase Diagram

Now, we construct the schematic $T-B$ phase diagram of the HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (see Figure 11). In Figure 11, the NFL region formed by the state FC is characterized by an excess of entropy $S_0$, defined by Equation (19). The border separating the AF state from the NFL region is shown by the solid curve $T_{NL}\left(B\right)$. At $B<B_{c0}$ and $T<T_{NL}\left(B\right)$, the system is in its AF state and exhibits LFL behavior [57]. The tricritical point $T_{\mathrm{cr}}$ at which the AF phase transition becomes of the first order is shown by the arrow. At that phase transition, the thermopower experiences the jump ${\mathrm{Jump}}_{\mathrm{F}}$ shown in Figure 6, changes its sign, and becomes $S/T>0$, for the hole states shown in Figure 4 are eliminated at $T<T_{\mathrm{cr}}$. At $T>T_{\mathrm{cr}}$, the AF transition is of the second order, and the thermopower experiences ${\mathrm{Jump}}_{\mathrm{S}}$ (see Figure 6) in the NFL region $S/T>0$, as seen in Figure 11. It is seen from the phase diagram Figure 11, that outside the area of the AF phase transition the $S_N=M_N^{*}$, considered a function of the dimensionless variable $T_N$ or $B_N$, has the universal scaling behavior. Figure 5, Figure 7a,b and Figure 8 demonstrate that all the data extracted from measurements on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, $\beta$-${\mathrm{YbAlB}}_4$, and $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$, collapse on the single scaling curve shown in Figure 5.

Here, it is worth noting a few remarkable points. As it follows from Figure 5 and Figure 7b, at $T_N<1$, the normalized thermopower ${(S/T)}_N$ becomes constant, thus exhibiting LFL behavior. However, at $T_N\simeq 1$, the system enters the crossover region. At growing temperatures and fixed magnetic field B, NFL behavior prevails. At low temperatures and growing B and at the tricritical point, the line of the second-order phase transitions changes to the line of the first-order phase transitions, as shown by the circle in Figure 11. The same change in the phase transition occurs in the archetypal HF metals $CeCoI{n}_5$ under the application of magnetic fields $B>B_{c2}\ge B_{c0}$, where $B_{c2}$ is the critical field destroying the superconducting state, and $B_{c0}$ is the critical field at which the topological FCQPT takes place [17,64]. This observation allows us to conclude that at lower temperatures and under the application of magnetic field B, the curve of any second-order phase transition passes into the curve of the first-order curve at the tricritical point $T_{\mathrm{cr}}$ (see Figure 11). Such a behavior occurs since, at $T_{\mathrm{cr}}$ the entropy $S_e$ starts to experience a jump because of the residual entropy $S_0$, occurring at the NFL state and given by Equation (19), is absent in the ordered state [15]. Thus, we again underline the important role of the magnetic field in the study of the properties of HF compounds.

7. Magnetic Field to Probe the Nature of Quantum Phase Transition

In this section, we show how to utilize magnetic field when probing a quantum phase transition that forms the properties of an HF metal. As an example, we consider the archetypal HF metal ${\mathrm{CeCoIn}}_5$.

The condensed-matter community is involved in active debate on the nature of the quantum phase transition (QPT) governing the low-temperature properties of heavy-fermion (HF) metals. Recent experimental observations of the archetypal HF metal ${\mathrm{CeCoIn}}_5$ at low temperatures probe the nature of its magnetic-field-tuned QPT under the superconducting state. These studies detected the smeared jump-like behavior revealed both in the residual resistivity ${\rho }_0$ and the Hall resistivity $R_H$, along with the violation of the time invariance symmetry $\mathcal{T}$ and the particle–hole symmetry $\mathcal{C}$. As a result, the studies provide vital clues on the nature of QPT taking place in ${\mathrm{CeCoIn}}_5$ (see, e.g., [28,64,65]). Based on a number of experimental data, we show that observations point out unambiguously that the QPT of ${\mathrm{CeCoIn}}_5$ is accompanied by the symmetry violation, and the QPT itself is represented by the topological FCQPT connecting two Fermi surfaces of different topological charges. We show that the behavior of ${\mathrm{CeCoIn}}_5$ at the FCQPT in magnetic fields and under pressure is related to the flattening of the single-particle spectrum induced by the FCQPT that exerts profound effects on ${\rho }_0$ and the differential tunneling conductance in the normal state. The application of a magnetic field or pressure to ${\mathrm{CeCoIn}}_5$ restores the Landau Fermi liquid behavior, leading to the suppression of both ${\rho }_0$ and the violation of $\mathcal{T}$ and $\mathcal{C}$ symmetries.

7.1. Residual Resistivity ${\rho }_0$

In the NFL state, the resistivity $\rho \left(T\right)$ is of the form

$$\rho \left(T\right)={\rho }_0+A{T}^n;n=1,$$

(22)

where ${\rho }_0$ is the residual resistivity, and A is a T-independent coefficient. The index n takes the values 1 and 2, respectively, for the NFL and LFL behaviors and $1\lesssim n\lesssim 2$ in the NFL-LFL transition (see Figure 5, Figure 12 and Figure 13). The residual resistivity ${\rho }_0$ usually results from the scattering of impurities. At $T>T_c$, the zero-field resistivity $\rho (T,B=0)$ varies linearly with T (see Equation (22)). At magnetic fields $B\ge B_{c2}$ and at low temperatures the resistivity exhibits LFL behavior, $\rho (T,B_{c2})\propto T^2$. Experimentally, ${\mathrm{CeCoIn}}_5$ is one of the purest heavy-fermion metals [65,66,67,68]. As a result, the regime of electron motion there is ballistic. Thus, under the application of the weak magnetic field B, one could observe a small positive contribution ${\delta }_B\propto B^2$ to ${\rho }_0$ arising from the orbital motion of electrons induced by the Lorentz force. As seen in Figure 12, this is not the case: specifically ${\rho }_0(B=B_{c0})\simeq 3.0$ μΩcm in the NFL state, while ${\rho }_0(B=6\mathrm{T})\simeq 0.3$ μΩcm in the LFL state (see the inset to Figure 12) [65,68]. Moreover, it is seen in Figure 3 that at elevated temperatures, all the resistivities taken at different fields $B=6,10,16$ T tend to coincide at the NFL state with $\rho \left(T\right)\propto T$ since the contribution ${\delta }_B$ is relatively small.

Another direct experimental confirmation of the change in ${\rho }_0$ is obtained in measurements on ${\mathrm{CeCoIn}}_5$ at various pressures P [69]. As seen in Figure 13, ${\rho }_0(P\to 0)\to 3.0$ μΩcm, and decreases by an order of magnitude to a value of about ${\rho }_0(P\to P^{*})\to 0.2$ μΩcm, with $P^{*}\simeq 1.6$ GPa [69]. Note that these values of the residual resistivity approximately coincide with those shown in Figure 12. Obviously, pressure P does not remove impurities from the sample. Thus, this large decrease in ${\rho }_0$ is due to the pressure-induced destruction of the FC state, as seen from the restoration of the LFL behavior at the same $P^{*}$. Similarly, the resistance $\rho (T,B)$ at fixed T as a function of B diminishes when the system transits from the NFL regime to the LFL regime under the application of magnetic field (the magnetoresistance becomes negative) [65]. This behavior is consistent with the FC theory [15]. Thus, the destruction of the FC state under the application of both magnetic fields B and pressure P entails a dramatic suppression of the flat band and its contribution to ${\rho }_0$ [32]. The same behavior of ${\rho }_0$ has been observed on twisted bilayer graphene, where a strong variation of ${\rho }_0$ is seen toward the magic angle [11,70]. In that case, the residual resistivity increases by more than three orders of magnitude and resembles the behavior of ${\rho }_0$ shown in Figure 13. These observations are explained with the framework of the FC theory [27,32], and, to our best knowledge, have not found any different explanations.

7.2. Asymmetric Tunneling Differential Conductivity

The particle–hole asymmetry is observed experimentally, analyzing the thermopower of HF compounds [43,44,45] (see Section 5). In this subsection, we continue to consider the particle–hole asymmetry taking place in HF compounds (see Section 4), and demonstrate that the differential tunneling conductivity (resistance) is an important instrument to explore the asymmetry in magnetic fields. The asymmetry takes place at a large scale of the universe [24], but here, we confine our consideration of HF compounds to illustrate the versatile and unique role of magnetic fields in analyzing the properties of HF compounds. Again, we shall see that the application of magnetic field makes the system transit from the NFL behavior to the LFL behavior (see Figure 2 and Figure 11).

Consider the differential tunneling conductivity ${\sigma }_d\left(V\right)$ between an HF metal and a simple metallic point. At low temperatures, ${\sigma }_d\left(V\right)$ can be noticeably asymmetric with respect to the change in voltage bias V, which makes the asymmetric part $\mathsf{\Delta }{\sigma }_d\left(V\right)={\sigma }_d\left(V\right)-{\sigma }_d(-V)$ finite. The asymmetry can be observed in experiments on HF metals whose electronic system has undergone the topological FCQPT, while the application of magnetic field causes the system to exhibit the LFL behavior and eliminates the asymmetry, as it has been predicted [24,29]. Such behavior has been observed in measurements on the HF metal ${\mathrm{CeCoIn}}_5$ [71], displayed in Figure 14. The data displayed in Figure 14 have been extracted from [71] (Figure S17) and demonstrate the sample independence of the asymmetry. As seen in Figure 14, $\mathsf{\Delta }{\sigma }_d\left(V\right)$ is a linear function of V and vanishes in the presence of magnetic field at a sufficiently low temperature of 1.75 K [15,24,29,30]. We note that at rising temperatures and fixed magnetic field B, the asymmetry emerges as soon as the NFL behavior again restores (see Figure 2a) since the elimination of the asymmetry is related to the crossover, rather than to a phase transition that takes place at finite temperature. We suggest that in the case of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ at QCP tuned by the application of magnetic field, the asymmetry exhibits a jump as it does in the Hall coefficient. In the case of ${\mathrm{CeCoIn}}_5$, such a jump is absent since the corresponding QCP is hidden under the superconducting dome (see Section 7.3).

Now, we analyze how the asymmetrical part $\mathsf{\Delta }{\sigma }_d\left(V\right)$ of the conductivity is related to the violation of $\mathcal{T}$ and $\mathcal{C}$ symmetries. Suppose we have a contact between HF and common metals. Let initially the electron current be directed from the HF metal to the usual metal. When voltage V is applied to the contact, we also change the electron charge $-e$ by $+e$, which changes the direction of the current. Consequently, one obtains exactly the above electric current under the voltage sign change $V\to -V$. As a result, the differential conductivity obtains the same asymmetric part $\mathsf{\Delta }{\sigma }_d\left(V\right)$, as seen in Figure 14. If $\mathcal{C}$ were conserved, the asymmetric part would vanish, $\mathsf{\Delta }{\sigma }_d\left(V\right)=0$. Thus, we conclude that because of the finite value of $\mathsf{\Delta }{\sigma }_d\left(V\right)$, the $\mathcal{C}$ symmetry is broken. In the same way, the time reversal $t\to -t$ for the constant charge generates the change in the current direction only. Since this reversal can also be achieved with $V\to -V$, it is clear that time-reversal symmetry is broken if $\mathsf{\Delta }{\sigma }_d\left(V\right)$ is finite. Hence, both the $\mathcal{C}$ and $\mathcal{T}$ symmetries are violated, provided that the non-zero $\mathsf{\Delta }{\sigma }_d\left(V\right)\ne 0$ emerges. In this case, the simultaneous transformation of $e\to -e$ and $t\to -t$ does not change anything, which means that the combined $\mathcal{CT}$ symmetry is conserved. It is worth noting that in the present case, the coordinates reversal symmetry, the parity $\mathcal{P}$, is not violated so that the combined general $\mathcal{CPT}$ symmetry is kept intact. It is well known that both $\mathcal{C}$ and $\mathcal{T}$ symmetries are preserved for systems of fermions described by the Landau theory. It follows that for such systems, as well as for ordinary metals, ${\sigma }_d\left(V\right)$ is a symmetric function of its variable V so that the conductivity asymmetry $\mathsf{\Delta }{\sigma }_d\left(V\right)$ is not observed in them at low T (see, e.g., [15,24,29]). Thus, $\mathsf{\Delta }{\sigma }_d\left(V\right)\ne 0$ signals the presence of FC and the corresponding flat band, which produce the violation of both $\mathcal{T}$ and $\mathcal{C}$, as explained in Ref. [24]. It is seen in Figure 15 that in ${\mathrm{CeCoIn}}_5$, in both superconducting and the pseudogap state, the asymmetric tunneling conductivity rises, $\mathsf{\Delta }{\sigma }_d\left(V\right)\ne 0$.

The data [72] displayed in Figure 15 are in good agreement with the data obtained in Ref. [73], as shown in Ref. [15]. These facts confirm that beyond $B_{c0}$, the FC state occurs, promoting both the superconducting and the corresponding pseudogap (PG) states [17]. As seen in Figure 15, at $T\le 2.7$ the asymmetric tunneling conductivity $\mathsf{\Delta }{\sigma }_d\left(V\right)$ is temperature independent. Such behavior is the intrinsic feature of HF metals located near the topological FCQPT [15]. This observation points to the fact that the PG state can be considered as the incoherent state emerging from the superconducting one. Thus, magnetic fields allow one to obtain unique information about the PG state.

We note that, as predicted by [15,20,24,29], $\mathcal{C}$-symmetry violation in strongly correlated Fermi systems [43,45] has been experimentally observed. It is seen from the phase diagram in Figure 2 that at raising magnetic fields B and low temperatures T, the HF metal ${\mathrm{CeCoIn}}_5$ exhibits LFL behavior and the symmetries are restored [24]. We note that the same behavior is exhibited by the HF metals ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ [74], ${\mathrm{YbCu}}_{5-\mathrm{x}}{\mathrm{Al}}_{\mathrm{x}}$ (for x = 1.5 [75]) and graphene [11], and explained within the framework of the FC theory [15]. We conclude that the phase transition in ${\mathrm{CeCoIn}}_5$ is represented by the topological FCQPT and accompanied by the symmetry violation, as it does in the mentioned above HF metals such as the archetypal HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (see Section 5) and graphene. We note that both $\mathcal{C}$ and $\mathcal{T}$ symmetries are violated due to the emergence of the flat band formed by the FC state (see Section 4). This observation is experimentally confirmed by the fact that the symmetries are restored under the application of magnetic field [15,24,29,75].

It is possible to measure the differential resistance $dV/dI$ as a function of the current I. In fact, the symmetry properties of this function are the same as those of ${\sigma }_d\left(V\right)$: when a magnetic field is applied, the asymmetry of the differential resistance disappears since the system goes into the LFL state. Magnetic field dependence of the differential resistance $dV/dI$ measured on graphene versus direct current I for different magnetic fields B is reported in Figure 16 [11]. Asymmetric part of the differential resistance $As\left(I\right)=dV/dI\left(I\right)-dV/dI(-I)$ decreases in an increased magnetic field and vanishes at $B\simeq 140$ mT, as can be seen in Figure 16. Such behavior is of great importance since the strongly correlated graphene has a perfect flat band [11]. Thus, in accordance with the prediction [15,29] the asymmetric part tends to zero at sufficiently high magnetic field, as seen in Figure 16. It is seen as well that the asymmetry remains at the superconducting state, and is suppressed at $B\simeq 140$ mT [24]. Thus, we can safely conclude that the strongly correlated state in graphene is controlled by the topological FCQPT, as it is in the HF compounds. To support our statement on the transition of graphene from the NFL behavior to the LFL, as it follows from vanishing the asymmetric part (see Figure 16), we predict that the resistance $\rho \left(T\right)$ versus T exhibits linear dependence, $\rho \left(T\right)\propto T$ in the normal state at zero magnetic field, while at relatively high magnetic fields and low temperatures of $k_{B}T<<{\mu }_{B}B$$\rho \left(T\right)$ exhibits $\rho \left(T\right)\propto T^2$ behavior, as it occurs in strongly correlated Fermi systems [15,16].

To conclude this subsection, we note that the violation of $\mathcal{C}$-symmetry in HF metals is directly sequestered with $\mathcal{T}$-symmetry violation. Meanwhile, the application of magnetic field B, driving the HF metal into its LFL state, restores both the $\mathcal{T}$ and $\mathcal{C}$ symmetries. One could suggest that the application of magnetic field is also to create the $\mathcal{T}$-symmetry violation. It is true, but the corresponding violation is rather weak in comparison with that induced by the presence of the flat band and needs special facilities to be observed. Again, we have to stress the valuable role of magnetic fields in exploring the properties of HF compounds.

7.3. Hall Effect

We are now in a position to consider a possible jump in the Hall coefficient of the ${\mathrm{CeCoIn}}_5$ at $B\to B_{c0}$ at the zero-temperature limit. Measurements of the Hall resistivity $R_H(T,B)$ in external magnetic fields B versus T have revealed a diverse low-temperature behavior of this basic property in HF metals, ranging from the LFL behavior to the challenging NFL one [28,31,33].

It is instructive to compare the behavior of the Hall coefficient in archetypal HF metals ${\mathrm{CeCoIn}}_5$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. At $T\to 0$ in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, the application of the critical magnetic field $B_{c0}$ suppressing the antiferromagnetic phase (with the Fermi momentum $p_{AF}\simeq p_F$) restores the LFL behavior with the Fermi momentum $p_f>p_F$, making an abrupt change in the Hall coefficient $R_H(T\to 0,B\to B_{c0})$, as a function of B [15,31,33]. At low temperatures and $B<B_{c0}$, the ground-state energy of the antiferromagnetic phase is lower than that of the heavy LFL, while at $B>B_{c0}$, the opposite happens, and the LFL state wins the competition. At $B=B_{c0}$, both the antiferromagnetic and the LFL states have the same ground-state energy. Thus, for $T=0$ and $B=B_{c0}$, an infinitesimal change in the magnetic field B leads to a finite jump in the Fermi momentum (see the phase diagram in Figure 11). In response to this change, the Hall coefficient $R_H\left(B\right)$ experiences a corresponding sharp jump [15,31,33]. In the case of ${\mathrm{CeCoIn}}_5$, one can hardly expect to observe such a jump since the QCP is hidden under the superconducting dome (see the phase diagram Figure 2a). As a result, one can only observe $R_H(B>B_{c0})$. Measurements of the Hall coefficient on ${\mathrm{CeCoIn}}_5$ do not show the possible jump, while the observed temperature dependence of $R_H(T,B)$ [28] can be explained within the framework of the FC theory [76]. The absence of the jump in the Hall coefficient could lead to the false impression that the quantum phase transition in ${\mathrm{CeCoIn}}_5$ is not accompanied by the symmetry violation, but the study of the $\mathcal{T}$ and $\mathcal{C}$ symmetries and the change in the residual resistivity ${\rho }_0$ clearly signals that the violation of the symmetry does take place.

8. Summary

In our short review, we have analyzed the influence of magnetic field on the properties of strongly correlated heavy-fermion compounds such as heavy-fermion metals and frustrated insulators with quantum spin liquid. We have shown that the magnetic field B can be considered a universal tool, allowing one to explore the main properties of HF compounds. These properties are the $T/B$ scaling, exhibiting under the application of magnetic field B at temperatures T and the emergence of the LFL behavior under the application of magnetic field. This emergence of the LFL behavior is accompanied by the restoration of the $\mathcal{T}$ and $\mathcal{C}$ symmetries and by the decrease of the residual resistivity ${\rho }_0$.

We have revealed and explained the universal scaling of the thermopower $S/T$ in such different HF compounds as ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, $\beta$-${\mathrm{YbAlB}}_4$, and $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_2\right]{\mathrm{CoO}}_2$. Our calculations are in good agreement with observations and show that the promoted universal $S/T$ scaling actually takes place. This scaling does not depend on the specific properties of the HF compounds under consideration and coincides with the scaling of the normalized effective mass $M_N^{*}={(C/T)}_N$, i.e., it is a scaling characteristic of HF compounds. We have also shown that destruction of the flattening of the single-particle spectrum profoundly affects $S/T$, eliminates the particle asymmetry, and leads to the two jumps and to the change of sign of the thermopower occurring at the antiferromagnetic phase transition.

We have delineated the quantum phase transition in ${\mathrm{CeCoIn}}_5$, and explained a number of experimental data collected in measurements on ${\mathrm{CeCoIn}}_5$ that allowed us to clarify the corresponding quantum phase transition. We have shown that the data related to the behavior under the application of magnetic fields of both the residual resistivity and the asymmetrical tunneling conductivity clearly point to the symmetry violation occurring at the quantum phase transition in ${\mathrm{CeCoIn}}_5$. From the experimental point of view, we have shown that these data demonstrate that the quantum phase transition in ${\mathrm{CeCoIn}}_5$ is accompanied by the symmetry breaking related to the violation of the $\mathcal{T}$ and $\mathcal{C}$ symmetries, and is represented by the topological FCQPT that connects two Fermi surfaces of different topological charges, as it occurs in graphene and the archetypal HF metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. We have explained why the Hall coefficient does not exhibit the jump at the phase transition in ${\mathrm{CeCoIn}}_5$, as it does in the case of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$.

Finally, our explanation of the number of experimental results within the framework of the fermion condensation theory suggests that the topological FCQPT is the intrinsic feature of many strongly correlated Fermi systems. This topological phase transition can be viewed as the universal cause for the non-Fermi liquid behavior and for the existence of the new state of matter represented by the strongly correlated Fermi systems (see, e.g., [15,16,17,24]).