Quantum Criticality in Heavy Fermion Metals

TL;DR

The paper surveys quantum criticality in heavy fermion metals, arguing that these systems realize two principal QCP classes: spin-density-wave (SDW) transitions driven by order-parameter fluctuations and local quantum critical points where Kondo screening collapses. It emphasizes that Kondo destruction introduces inherently quantum excitations and a Fermi-surface reconstruction, leading to multiple energy scales and non-Fermi-liquid behavior that extend over broad regions of the phase diagram. Thermodynamic and transport measurements, including divergent Grüneisen ratios, $ ext{ω}/T$ scaling, and abrupt Hall coefficient changes, support a picture where both mass divergence and Fermi-surface collapse accompany the QCPs. The work also discusses the interplay between quantum criticality and unconventional superconductivity, the variation across materials, and the need for a unified theoretical framework that can capture both Kondo-driven and SDW-driven criticalities and their implications for broader strongly correlated systems.

Abstract

Quantum criticality describes the collective fluctuations of matter undergoing a second-order phase transition at zero temperature. Heavy fermion metals have in recent years emerged as prototypical systems to study quantum critical points. There have been considerable efforts, both experimental and theoretical, which use these magnetic systems to address problems that are central to the broad understanding of strongly correlated quantum matter. Here, we summarize some of the basic issues, including i) the extent to which the quantum criticality in heavy fermion metals goes beyond the standard theory of order-parameter fluctuations, ii) the nature of the Kondo effect in the quantum critical regime, iii) the non-Fermi liquid phenomena that accompany quantum criticality, and iv) the interplay between quantum criticality and unconventional superconductivity.

Figures (8)

• Figure 1: Quantum critical points in heavy fermion metals. a: AF ordering temperature $T_N$ vs. Au concentration $x$ for CeCu$_{6-x}$Au$_x$ (Ref. Schroeder), showing a doping induced QCP. b: Suppression of the magnetic ordering in YbRh$_2$Si$_2$ by a magnetic field. Also shown is the evolution of the exponent $\alpha$ in $\Delta\rho \equiv [\rho(T)-\rho_0 ]\propto T^\alpha$, within the temperature-field phase diagram of YbRh$_2$Si$_2$ (Ref. Custers). Blue and orange regions mark $\alpha=2$ and $1$, respectively. c: Linear temperature dependence of the electrical resistivity for Ge-doped YbRh$_2$Si$_2$ over three decades of temperature (Ref. Custers), demonstrating the robustness of the non-Fermi liquid behavior in the quantum critical regime. d: Temperature vs. pressure phase diagram for CePd$_2$Si$_2$, illustrating the emergence of a superconducting phase centered around the QCP. The Néel- ($T_N$) and superconducting ordering temperatures ($T_c$) are indicated by closed and open symbols, respectively. Mathur
• Figure 2: Schematic phase diagrams displaying two classes of quantum critical points. Shown are the temperature/energy scales vs. control-parameter ($\delta$, which tunes the ratio of the Kondo interaction to the RKKY interaction), illustrating quantum criticality with critical Kondo destruction (a) and of the spin-density-wave type (b). $T_N$ represents the Néel temperature and $T_{\text{\small FL}}$ the onset of the low-temperature Fermi liquid regime. Lines with arrow correspond to renormalization-group flows, which describe the transformation of the system from the high temperature fully-incoherent regime to the zero-temperature ground states. $E_{\rm loc}^*$ marks an energy scale separating the renormalization-group flows towards two types of ground states -- one with a large Fermi surface (Kondo resonance fully developed, and $f$-electrons delocalized) and the other with a "small" Fermi surface (static Kondo screening absent, and $f$-electrons localized). Similar renormalization-group flows apply to (b), but are omitted there for visual simplicity. $T_0$ signifies the initial crossover in a Kondo lattice system, from the high temperature regime, where the local moments are completely incoherent, to the intermediate temperature regime in which the initial Kondo screening operates.
• Figure 3: Divergence of the dimensionless critical Grüneisen ratio at quantum critical points. Plotted are the temperature dependence of $\Gamma^{cr}=V_m/\kappa_T\times \beta^{cr}/C^{cr}$ for CeNi$_2$Ge$_2$ (a) and YbRh$_2$(Si$_{0.95}$Ge$_{0.05}$)$_2$ (b), on double-log scales.Kuechler 2003 Here, $\beta^{cr}$ and $C^{cr}$ are the volume thermal expansion and specific heat after subtraction of normal (Fermi-liquid) contributions. In addition, $V_m$ and $\kappa_T$ are the molar volume and isothermal compressibility (at room temperature); they are used as normalization to make $\Gamma^{cr}$ dimensionless. Error bars, standard errors.
• Figure 4: Thermodynamic and transport properties close to the quantum critical point in YbRh$_2$Si$_2$. a: Low-temperature electronic specific-heat coefficient at various different fields applied perpendicular to the $c$-axis [Oeschler, N. et al.Physica B, in press (2007)]. At the critical field, $H_c \approx 0.055$ T, the specific-heat coefficient goes as $T^{-0.4}$ at low temperatures and as $-{\rm ln}T$ at higher temperatures. b: The zero-field data over an extended temperature range, illustrating the onset of the ${\rm ln}T_0/T$ dependence at $T \approx 10$ K ($T_0 = 24$ K) and the onset of a power-law divergence at $T \approx 0.3$ K. Trovarelli c: Low-temperature electrical resistivity [Gegenwart, P. et al.Physica B, in press (2007)] as a function of temperature at the same magnetic fields as in a.
• Figure 5: Evidence for an additional low-energy scale in the Hall effect and thermodynamic data of YbRh$_2$Si$_2$. a: Linear-response Hall coefficient $R_H$ as derived from the initial slope of the Hall resistivity in a crossed field experiment [Friedemann, S. et al.Physica B, in press (2007)], performed on the same single crystal used in Ref. Paschen Error bars, standard errors. The crossover width decreases with temperature, extrapolating to zero in the zero-temperature limit.Paschen b: Magnetic-field dependence of the magnetostriction of a high-quality single crystal ($\rho_0 \approx 0.5~\mu\Omega$cm). The symbols represent the linear coefficient $\lambda_{[110]}=\partial\ln L/\partial H$ (where $L$ is the sample length along the $[110]$ direction within the tetragonal $ab$ plane) versus $H$ at different temperatures.Gegenwart 2007