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

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## Abstract

**:**

## 1. Introduction

^{3}He 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].

^{3}He 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

^{3}He or by frustrated insulators with quantum spin liquid, which have nothing to do with fluctuations, the spin-density-wave scenario, or the Kondo effect.

## 2. The Effective Mass M*

## 3. The Scaling of the Effective Mass

## 4. Flat Bands and Particle–Hole Asymmetry

## 5. Thermopower under the Application of Magnetic Fields

#### 5.1. Scaling

**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.

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

_{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 YbRh_{2}Si_{2}

**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.

#### Schematic $T-B$ Phase Diagram

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

#### 7.1. Residual Resistivity ${\rho}_{0}$

#### 7.2. Asymmetric Tunneling Differential Conductivity

#### 7.3. Hall Effect

## 8. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Universal scaling behavior of the normalized specific heat ${(C/T)}_{N}$ as a function of the normalized temperature ${T}_{N}$. ${(C/T)}_{N}$ is obtained from measurements of the heat capacity $C/T$ at ${\mathrm{YbRh}}_{2}{\mathrm{Si}}_{2}$ in magnetic fields B [38] depicted in the legend. The LFL region, crossover region, and NFL region are displayed by the arrows. The solid curve represents our calculations of ${(C/T)}_{N}={M}_{N}^{*}$ based on Equations (10) and (15) (see Figure 2b) [15].

**Figure 2.**Panel (

**a**): Schematic phase diagram of HF metals. ${B}_{c0}$ is magnetic field at which the effective mass diverges. $\mathrm{SCAF}$ denote the superconducting (SC) and antiferromagnetic (AFM) states, respectively. At $B\lesssim {B}_{c0}$, the system can be captured by AFM or SC states. The vertical arrow depicts the transition from the LFL regime to the NFL regime at fixed B. The horizontal arrow illustrates the HF metals in question transiting from the NFL to the LFL regime along B at fixed T. Panel (

**b**) displays the schematic plot of the universal scaling behavior of the normalized effective mass ${M}_{N}^{*}$ versus the normalized temperature ${T}_{N}\propto {(T/B)}_{N}$. The crossover region, where ${M}_{N}^{*}$ reaches its maximum value ${M}_{M}^{*}$ at $T={T}_{M}$, is shown by the hatched area. The system transits from the NFL to the LFL behavior at rising B at fixed T, which is at $T/B<1$. The arrows mark the LFL and the NFL regions and the transition region in the behavior of ${M}_{N}^{*}$ as a function of ${T}_{N}={(T/B)}_{N}$. The NFL behavior is characterized by ${M}^{*}\propto {(T/B)}^{-2/3}\simeq {(T/B)}^{-0.66}$ (see Equation (12)).

**Figure 3.**Universal $B/T$ scaling of thermodynamic properties of strongly correlated Fermi systems. Panel (

**a**): Scaling of magnetic susceptibility $\chi {T}^{2/3}$ of the HF metal ${\mathrm{CeCu}}_{6-\mathrm{x}}{\mathrm{Au}}_{\mathrm{x}}$. Data are extracted from experimental measurements [40], and that of ${\mathrm{ZnCu}}_{3}{\left(\mathrm{OH}\right)}_{6}{\mathrm{Cl}}_{2}$ from data [41]. At $B/T\ll 1$ m the systems exhibit NFL behavior with $\chi \propto {M}^{*}$, as given by Equation (11), i.e., ${T}^{2/3}\chi \propto $ const. At $B/T\gg 1$, the systems demonstrate LFL behavior, with $\chi $ as given by Equation (12), being a decreasing function of $B/T$ (see Equation (15)). Panel (

**b**): $T/B$ scaling of the specific heat ${C}_{mag}/T$ of ${\mathrm{ZnCu}}_{3}{\left(\mathrm{OH}\right)}_{6}{\mathrm{Cl}}_{2}$ is extracted from data [42]. At $T/B\ll 1$, the systems demonstrate LFL behavior with $\chi \propto const$. At $T/B>1$, the systems exhibit NFL behavior with $\chi \propto {T}^{2/3}$.

**Figure 4.**The single-particle energy $\epsilon \left(\mathbf{p}\right)$ and the distribution function $n\left(\mathbf{p}\right)$ at $T=0$. The arrow shows the Fermi energy ${E}_{F}=\mu $. The vertical lines denote the FC area ${p}_{i}<p<{p}_{f}$ with $0<{n}_{0}\left(p\right)<1$ and $\epsilon \left(\mathbf{p}\right)={E}_{F}$. The Fermi momentum ${p}_{F}$ is in the interval ${p}_{i}<{p}_{F}<{p}_{f}$ and corresponds to the Landau Fermi liquid, emerging when the FC state and the corresponding particle–hole asymmetry are eliminated. The single-particle energy $\epsilon \left(\mathbf{p}\right)$ and distribution function $n\left(\mathbf{p}\right)$ of the LFL state are displayed by the blue lines. The arrow indicates the hole states generated by the flat band.

**Figure 7.**Universal scaling behavior of the normalized ${(S/T)}_{N}$. (

**a**) Normalized isotherm ${(S\left(B\right)/T)}_{N}$ as a function of normalized magnetic field ${B}_{N}$ at different temperatures T displayed in the legend. Outside the antiferromagnetic phase, the data show the universal scaling behavior. (

**b**) The normalized thermopower ${(S/T)}_{N}$ in magnetic fields B depicted in the legend. The experimental data are extracted from measurements on ${\mathrm{YbRh}}_{2}{\mathrm{Si}}_{2}$ [46,47] and on $\beta $-${\mathrm{YbAlB}}_{4}$ [59]. The data, taken at the AF phase [46,47] and at the superconducting one (SC) [59] (delineated by the ellipse and the rectangle, respectively) expose the violation of the scaling. The theoretical solid curves in (

**a**,

**b**) coincide with that shown in Figure 2b and Figure 5 [15,54].

**Figure 8.**Scaling of $S/T$. At magnetic field $B=0$, the strongly correlated layered cobalt oxide $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_{2}\right]{\mathrm{CoO}}_{2}$ demonstrates the scaling of ${(S\left(T\right)/T)}_{N}$ versus ${T}_{N}$. The data are extracted from measurements on $\left[{\mathrm{BiBa}}_{0.66}{\mathrm{K}}_{0.36}{\mathrm{O}}_{2}\right]{\mathrm{CoO}}_{2}$ [56]. The solid curve displaying the theoretical calculations is the same as that shown in Figure 5.

**Figure 10.**The temperature dependence of the ratios ${\gamma}_{norm}=({\gamma}_{0}-{A}_{1})/{B}_{1}$ for $t<1$ and $t>1$ as a function of $|1-t|$ given by Equation (21) is shown by the solid line. The ratios are extracted from the data obtained in measurements of ${\gamma}_{0}$ on ${\mathrm{YbRh}}_{2}{\mathrm{Si}}_{2}$ at the AF phase transition [63] and are shown as triangles, as shown in the legend.

**Figure 11.**Schematic $T-B$ phase diagram of ${\mathrm{YbRh}}_{2}{\mathrm{Si}}_{2}$. Vertical and horizontal arrows crossing the transition region marked with thick lines show LFL-NFL and NFL-LFL transitions at fixed B and T, respectively. The hatched area represents the crossover between the NFL and LFL regimes. As shown by the solid curve, at $B<{B}_{c0}$, the system is in the AF state and exhibits LFL behavior [57]. The AF phase transition line is designated ${T}_{NL}\left(B\right)$. The tricritical point marked with the arrow is at the point $T={T}_{\mathrm{cr}}$. At $T<{T}_{\mathrm{cr}}$ the AF phase transition becomes of the first order, and is depicted by the orange dots.

**Figure 12.**Resistivity $\rho (T,B)$ obtained in measurements on ${\mathrm{CeCoIn}}_{5}$ under the application of magnetic fields B displayed in the legend [65]. The inset (

**a**,

**b**) shows both the LFL behavior of the resistivity at low temperatures and the crossover with $1\lesssim n\lesssim 2$ at elevated T.

**Figure 13.**Adapted from [69]. Values of the residual resistivity ${\rho}_{0}$ (left axis, solid squares) and the index n in the fit $\rho \left(T\right)={\rho}_{0}+A{T}^{n}$ (right axis, solid squares) versus pressure P.

**Figure 14.**Asymmetric part $\mathsf{\Delta}{\sigma}_{d}\left(V\right)$ of the tunneling differential conductivity measured on ${\mathrm{CeCoIn}}_{5}$ and extracted from the experimental data [71]. Linear dependence of $\mathsf{\Delta}{\sigma}_{d}$ is shown by the straight line. The asymmetric part disappears at $B=14$ T and $T=1.75$ K, with ${B}_{c0}\simeq 5$ T.

**Figure 15.**The asymmetric part of tunneling conductivity $\mathsf{\Delta}{\sigma}_{d}\left(V\right)$ in ${\mathrm{CeCoIn}}_{5}$, extracted from the experimental data [72]. At $T\le 2.7$ K ${\mathrm{CeCoIn}}_{5}$ is in its pseudogap (PG) and superconducting states [72]. At $T\le 2.7$ K, as it is shown by both the ring and the arrow, $\mathsf{\Delta}{\sigma}_{d}\left(V\right)$ is temperature independent [15,17].

**Figure 16.**Magnetic field dependence of the asymmetric part $dV/dI\left(I\right)-dV/dI(-I)$ on the current I extracted from [11]. The different values of B are shown in the inset.

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**MDPI and ACS Style**

Shaginyan, V.R.; Msezane, A.Z.; Artamonov, S.A.
Magnetic Field as an Important Tool in Exploring the Strongly Correlated Fermi Systems and Their Particle–Hole and Time-Reversal Asymmetries. *Magnetism* **2023**, *3*, 180-203.
https://doi.org/10.3390/magnetism3030015

**AMA Style**

Shaginyan VR, Msezane AZ, Artamonov SA.
Magnetic Field as an Important Tool in Exploring the Strongly Correlated Fermi Systems and Their Particle–Hole and Time-Reversal Asymmetries. *Magnetism*. 2023; 3(3):180-203.
https://doi.org/10.3390/magnetism3030015

**Chicago/Turabian Style**

Shaginyan, Vasily R., Alfred Z. Msezane, and Stanislav A. Artamonov.
2023. "Magnetic Field as an Important Tool in Exploring the Strongly Correlated Fermi Systems and Their Particle–Hole and Time-Reversal Asymmetries" *Magnetism* 3, no. 3: 180-203.
https://doi.org/10.3390/magnetism3030015