Critical fluctuations and conserved dynamics in a strange ferromagnetic metal

Abstract

The origin of the strange metallic behavior observed in a wide range of quantum materials is an open challenge to condensed matter physics. Historically, strange metals were uniquely associated with antiferromagnetic quantum critical points (QCPs), but a new generation of materials reveals their association with uniform order parameters, such as ferromagnetism, valley or nematic order, suggesting a deeper common denominator. At a QCP, order parameter fluctuations are characterized by the dynamical critical exponent $z$, which quantifies the space-time scaling asymmetry. Here, we report the observation of a divergence in the Grüneisen ratio at the QCP of the strange-metal ferromagnet CeRh$_6$Ge$_4$ with a dynamical critical exponent $z=3$, signaling that the underlying quantum singularity involves a conserved degree of freedom. Yet the magnetization of this easy-plane ferromagnet is not conserved. We argue that the $z=3$ strange criticality requires a description beyond the Landau paradigm, proposing a link with the gauge modes of the small-to-large Fermi surface transition and the associated gauge charge of the delocalizing heavy electrons.

Figures (4)

• Figure 1: Temperature dependence of the resistivity of Ce(Rh$_{1-x}$Co$_x$)$_6$Ge$_4$ with the current along the $c$ axis for (a) $x\leq0.021$ and (b) $x\geq0.045$. The ferromagnetic transitions at $T_{\rm C}$ are detected in the $x=0$ and $x=0.021$ samples as shown by the vertical arrows, which for the resistivity are clearly observed in the derivative in the inset of (a). $\rho(T)$ for $x=0$ is shifted vertically by $3.6~\mu\Omega~{\rm cm}$ for clarity. The dashed lines show the $\Delta\rho\sim T^2$ behavior present below $T_{\rm FL}$. Temperature dependence of the (c) electronic specific heat coefficient $C_e/T$ (after subtracting the data of LaRh$_6$Ge$_4$), of Ce(Rh$_{1-x}$Co$_x$)$_6$Ge$_4$ single crystals down to 0.06 K, and (d) the real part of the ac susceptibility $\chi'$, where the ferromagnetic transition corresponds to a peak marked by vertical arrows. The inset of (c) magnifies the low temperature $C_e/T$ below 0.5 K.
• Figure 2: (Color online) Temperature dependence of the linear thermal expansion coefficient (a) perpendicular ($\alpha_{\perp c}$), and (b) parallel ($\alpha_{\parallel c}$), to the $c$-axis. For samples exhibiting magnetic transitions, there is a drop of $\alpha/T$ upon cooling for both directions. For $x=0.050$, 0.055, and 0.066, $\alpha/T$ continues to increase with decreasing temperature, which can be described by a power law behavior $\alpha/T\sim T^{-n}$ as shown by the solid lines, where the corresponding $n$ for $\alpha_{\perp c}$ are 0.70(6), 0.60(4), and 0.61(3), respectively, while for $\alpha_{\parallel c}$ they are 0.70(6), 0.47(6), and 0.33(5).
• Figure 3: Temperature-doping phase diagram of Ce(Rh$_{1-x}$Co$_x$)$_6$Ge$_4$ determined from resistivity, ac susceptibility, specific heat, and thermal expansion measurements. The $T_{\rm C}$ for the specific heat and thermal expansion are determined from the midpoints and change of sign of $\beta(T)$, respectively, while the error bars correspond to the transition widths. The ferromagnetic transition is suppressed by doping, and is no longer detected for $x=0.050$, which is therefore in close proximity to a ferromagnetic QCP.
• Figure 4: (a) Temperature dependence of the specific heat coefficient (left axis) and resistivity (right axis) of a sample close to the critical concentration, Ce(Rh$_{0.95}$Co$_{0.05}$)$_6$Ge$_4$, measured down to 0.06 K. Clear strange metal behavior is observed, with the resistivity showing a linear-temperature dependence, while $C_e/T$ continues to increase with decreasing temperature, following a logarithmic temperature dependence (red line), which exhibits a slope change below around 0.5 K. (b) Temperature dependence of the volume thermal expansion (as $\beta/T$), together with fit to a power law $\beta/T\sim T^{-n}$ shown by the solid line with $n=0.67(4)$. The inset shows the temperature dependence of the Grüneisen ratio $\beta/C_e$ that diverges at low temperatures, with an exponent $n=0.66(9)$.