Colloquium: Heavy-electron quantum criticality and single-particle spectroscopy

Angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) have become indispensable tools in the study of correlated quantum materials. Both probe complementary aspects of the single-particle excitation spectrum. Taken together, ARPES and STM have the potential to explore properties of the electronic Green’s function, a central object of many-body theory. This review explicates this potential with a focus on heavy-electron quantum criticality, especially the role of Kondo destruction. A discussion on how to probe the Kondo destruction effect across the quantum-critical point using ARPES and STM measurements is presented. Particular emphasis is placed on the question of how to distinguish between the signatures of the initial onset of hybridization-gap formation, which is the “high-energy” physics to be expected in all heavy-electron systems, and those of Kondo destruction, which characterizes the low-energy physics and, hence, the nature of quantum criticality. Recent progress and possible challenges in the experimental investigations are surveyed, the STM and ARPES spectra for several quantum-critical heavy-electron compounds are compared, and the prospects for further advances are outlined.

Figure 1

The quantum-critical fan. (a) A continuous quantum phase transition occurs at zero temperature for a critical value (${\delta }_c$) of a nonthermal tuning parameter $\delta$. It separates the distinct behaviors of the ground state wave function, i.e., being ordered for $\delta <{\delta }_c$ (orange line) and disordered for $\delta >{\delta }_c$. At nonzero temperatures vestiges of the quantum phase transition lead to distinctive scaling behavior in a quantum-critical fan that spreads out of the quantum-critical point and extends up to a problem specific cutoff temperature. Unlike the scaling behavior, the existence of a fan of quantum-critical behavior is generic, independent of the nature of the criticality. (b) Frequently, quantum phase transitions occur as order is suppressed and the transition temperature $T_c(\delta )$ of a classical phase transition vanishes, i.e., $T_c(\delta \to {\delta }_c)\to 0$. In heavy-electron materials, the most common types of quantum criticality separate antiferromagnetic and paramagnetic phases. Shown here is a type of antiferromagnetic order, indicated by the variation of the magnetic moment density over one wavelength. This Colloquium explores the potential of single-particle spectroscopies to distinguish different types of quantum criticality.

Figure 2

Basic concepts of Kondo destruction quantum criticality. (a) Local quantum criticality with Kondo destruction under the variation of the control parameter $\delta$. Here $T_0$ is a high-energy scale that describes the initial onset of dynamical Kondo correlations and that smoothly evolves across the QCP ${\delta }_c$. This high-energy scale is reflected in the onset of hybridization-gap formation. The low-energy physics is described in terms of $T_N$ and $T_{\mathrm{FL}}$, which are, respectively, the temperatures for the Néel transition and the crossover into the paramagnetic Fermi liquid state. This phase diagram also involves the Kondo destruction energy scale $E_{\mathrm{loc}}^{*}$, which characterizes the Kondo destruction. The $E_{\mathrm{loc}}^{*}$ line divides the phase diagram in terms of the flow of the system toward either the Kondo screened or the Kondo destruction ground state. In the conventional model of spin-density-wave quantum criticality, the line $E_{\mathrm{loc}}^{*}$ extrapolates to zero temperature in the ordered phase so that the Fermi surface already is large before reaching the QCP with increasing $\delta$ and evolves smoothly across the QCP ([19]; [121]). Adapted from [128]. (b) The small (left) and large (right) Fermi surfaces and the associated quasiparticle weights $z_S$ and $z_L$ that are discussed in Sec. 3. The fluctuating Fermi surfaces (middle) are associated with the QCP. Adapted from [96].

Figure 3

Sketch of the optical conductivity $\sigma (\omega )$ for temperatures well above (dashed black line) and well below (continuous red line) the crossover temperature scale $T_0$. Here the lowering of temperature through $T_0$ is accompanied by the onset of the hybridization gap. The characteristic frequency scale for the hybridization gap is $\sqrt{T_{0}D}$ (the vertical dotted blue line), where $D$ is an energy scale of the order of the conduction electron bandwidth. At low energies, i.e., for $\omega \ll T_0$ and $T\ll T_0$, and sufficiently far away from quantum criticality (i.e. $\delta <{\delta }_c$ or $\delta >{\delta }_c$), a pronounced Drude peak reflects the mass enhancement in the Fermi liquid regimes that surround the QCP in the phase diagram; see Fig. 2. The behavior of $\sigma (\omega )$ at high energies, including the hybridization gap, is a generic feature of heavy-electron systems and is seen throughout the high-energy part of the phase diagram of Fig. 2.

Figure 4

Electronic characteristics of the Fermi liquid state of a Kondo lattice. (a) Spectral density of a Fermi liquid: The quasiparticle pole at $E_{\mathbf{k}}$ has a characteristic width $\delta E$ that increases with the distance from the Fermi energy $E_F$ as $\delta E\sim |E_{\mathbf{k}}-E_F{|}^2$. A similar broadening occurs due to finite temperature effects. The incoherent part of $A(E,\mathbf{k})$ vanishes at $E_F$. (b) Quasiparticle dispersion in the Fermi liquid to either side of the QCP: $k_F^L$ and $k_F^S$ refer to large and small Fermi surfaces, respectively. Across a Kondo destruction QCP, the one-electron spectrum is gapless at $k_F^L$ and develops a small gap at $k_F^S$ for $\delta >{\delta }_c$, and the converse is valid for $\delta <{\delta }_c$. The flattening of the dispersion near $k_F^S$ for $\delta <{\delta }_c$ (dashed blue curve) reflects the effective mass enhancement due to the dynamical Kondo effect.

Figure 5

Quantum criticality in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. (a) The temperature vs field phase diagram of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. The blue regions mark Fermi liquid behavior, i.e., $\rho (T)-\rho (0)\sim T^2$, while orange indicates the quantum-critical area of the phase diagram where $\rho (T)-\rho (0)\sim T^x$, with $x\approx 1$. The continuous line in the quantum-critical region is the $E^{*}$ line as derived from thermodynamic and transport properties ([90]; [41]; [34]). Adapted from [24]. (b) Normalized Hall coefficient across the critical field for different temperatures. The inverse of $R_H$ is a measure of the carrier density. The lower $T$, the sharper is the crossover. At $T=0$ and $B=B_c$, a jump of $R_H$ corresponds to the sudden localization of $4f$ electrons as $B$ is taken trough $B_c$ from above. From [90]. (c) Comparison between the isothermal magnetotransport crossover width and the crossover field as specified by the ratio of the $\mathrm{FWHM}/2$ to the crossover inflection field $B_{\inf }$. FWHM denotes the full width at half maximum. From [89]. (d) The “sharpness” of the crossover: The FWHM vanishes in a linear-in-$T$ fashion indicating a jump of $R_H$ at $B_c$ in the zero-temperature limit. From [34].

Figure 6

Divergence of the $T^2$ coefficient $A$ of the resistivity at the QCP in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. Through the Kadowaki-Woods relation, this implies that the effective mass diverges on approach to the critical field $B_c={\mu }_0{H}_c$ from either side of the QCP. Data for $H\perp c$ have been scaled by a factor of 11. Adapted from [38].

Figure 7

STM spectroscopy of the lattice Kondo feature at $-6\mathrm{meV}$ in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. (a) The temperature evolution of the peak height of the $-6\mathrm{meV}$ peak. A strong increase in the peak height is observed below 5 K. (b) The FWHM of the $-6\mathrm{meV}$ peak across the critical field at the base temperature $T=0.3\mathrm{K}$. Note that at this temperature all field values place the system within the quantum-critical fan. The decrease of the peak width near $B^{*}={\mu }_0{H}^{*}(T=0.3\mathrm{K})$ is consistent with a critical slowing down at quantum criticality. From [111].

Figure 8

Temperature dependence of the resistivity $\rho$ of ${\mathrm{CeCoIn}}_5$. (a) $\rho (T)$ in the temperature range from 2.2 to 300 K. $\rho$ has a smooth maximum around $T_{\mathrm{coh}}\approx 40\mathrm{K}$; this temperature for the resistivity maximum is commonly referred to as the coherence temperature, which is a manifestation of $T_0$ defined earlier. (b) $\rho$ (black continuous line) and the magnetic resistivity ${\rho }_m$ (red continuous line) in a semilog plot for temperatures from 2 to 200 K. ${\rho }_m(T)$ is defined as the difference between the resistivity of ${\mathrm{CeCoIn}}_5$ and that of its nonmagnetic reference compound ${\mathrm{LaCoIn}}_5$ at temperature $T$. The dashed line represents a linear law fit to ${\rho }_m(T)$ and shows that ${\rho }_m(T)$ is linear in $T$ from $T_c$ to approximately 20 K.

Figure 9

ARPES view of the $4f$ electron weight near the Fermi energy in ${\mathrm{CeCoIn}}_5$. (a) Evidence for the initial development of hybridization between $4f$ and conduction band $\alpha$ at $T=60$ and $17\mathrm{K}$ after dividing the EDCs by the Fermi-Dirac function. (b) $\omega /T$ scaling of the EDCs near the $\mathrm{\Gamma }$ point in an intermediate temperature range and energy range around the Fermi energy $E_F$. (c) Background subtracted $4f$ electron spectral weight transfer near the $\mathrm{\Gamma }$ point vs temperature. The EDCs have been integrated over an energy window from $-40$ to 2 meV. From [14].

Figure 10

Tunneling spectroscopy of ${\mathrm{CeCoIn}}_5$ and ${\mathrm{CeRhIn}}_5$: local conductance vs applied bias voltage for different temperatures on (a) Ce-terminated surfaces and (b) Co-(respectively, Rh-) terminated surfaces. The peak-dip-peak structure in conductance of ${\mathrm{CeCoIn}}_5$ (a) is typical of a hybridization gap that is not obvious in ${\mathrm{CeRhIn}}_5$, even at the lowest temperature. From ([4]).

Figure 11

de Haas–van Alphen measurements on ${\mathrm{CeRhIn}}_5$. (a) Jump of the dHvA frequencies at $p_c$ indicating a reconstruction of the Fermi surface as the QCP is crossed. (b) Diverging effective mass upon approaching $p_c$ from above and below. From [115].

Figure 12

EDCs of ${\mathrm{CeRhIn}}_5$ vs temperature. The energy-distribution curves show the evolution of spectral weight with temperature near the Fermi energy $E_F$ for the three bands that cross $E_F$, labeled $\alpha$, $\beta$, and $\gamma$. Data were taken along the $\mathrm{\Gamma }\mathrm{M}$ direction at $k_{\parallel }=-0.57{\AA }^{-1}$ ($\alpha$ band), $k_{\parallel }=-0.3{\AA }^{-1}$ ($\beta$ band), and $k_{\parallel }=-0.124{\AA }^{-1}$ ($\gamma$ band) and with an uncertainty of $\delta k_{\parallel }\sim 0.03{\AA }^{-1}$ for each of the three $k_{\parallel }$ values. From [16].

Figure 13

Optical conductivity $\sigma (\omega ,T)$ and evolution of the hybridization gap. (a) Although the hybridization gap in ${\mathrm{CeRhIn}}_5$ (top) is overall less pronounced than that in the optical conductivity of ${\mathrm{CeCoIn}}_5$ (bottom), the overall features for both compounds are in accordance with general expectations (see Fig. 3): at the highest measured $T$ a broad Drude peak exists out of which a hybridization gap develops below $\sqrt{T_{0}D}$ as $T$ is lowered. From [77]. (b) The hybridization gap in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ evolves over a large $T$ region, starting well above 100 K. As the data are taken at zero external field, the system is located on the $\delta <{\delta }_c$ side (see Sec. 2) and a Drude peak is therefore expected in $\sigma (\omega ,T)$ at small $\omega$ and sufficiently low $T$. From [55]. (c) In ${\mathrm{CeCu}}_6$ the optical conductivity develops a hybridization gap at around $\hslash \omega \approx 1\mathrm{meV}$ below 50 K which is flanked toward higher energies by a pronounced peak. From [74].