Nonvanishing Energy Scales at the Quantum Critical Point of ${\mathrm{CeCoIn}}_5$

Heat and charge transport were used to probe the magnetic field-tuned quantum critical point in the heavy-fermion metal ${\mathrm{CeCoIn}}_5$. A comparison of electrical and thermal resistivities reveals three characteristic energy scales. A Fermi-liquid regime is observed below $T_{\mathrm{FL}}$, with both transport coefficients diverging in parallel and $T_{\mathrm{FL}}\to 0$ as $H\to H_c$, the critical field. The characteristic temperature of antiferromagnetic spin fluctuations, $T_{\mathrm{SF}}$, is tuned to a minimum but finite value at $H_c$, which coincides with the end of the $T$-linear regime in the electrical resistivity. A third temperature scale, $T_{\mathrm{QP}}$, signals the formation of quasiparticles, as fermions of charge $e$ obeying the Wiedemann-Franz law. Unlike $T_{\mathrm{FL}}$, it remains finite at $H_c$, so that the integrity of quasiparticles is preserved, even though the standard signature of Fermi-liquid theory fails.

Figure 1

Thermal conductivity of ${\mathrm{CeCoIn}}_5$, plotted as $\kappa /T$ vs $T$ (main panel) and as electronic [17] thermal resistivity $w=L_{0}T/{\kappa }_e$ vs $T^2$ (left inset), for $H\parallel [001]$. The data in the left inset, offset for clarity, are for $H=6$, 7, 8, 10, and 12 T (bottom to top); lines are linear fits valid up to $T=T_{\mathrm{FL}}$, the Fermi-liquid temperature, marked by an arrow for $H=10\mathrm{T}$. Right inset: field dependence of the $T^2$ Fermi-liquid coefficients of charge and heat transport.

Figure 2

Upper panels: comparison of thermal [$w(T)$; open symbols] and electrical [$\rho (T)$; solid symbols] resistivities at the critical field (5.25 T), plotted vs $T^{3/2}$ (left), and in the FL regime (at 10 T), plotted vs $T^2$ (right). Lower panel: normalized Lorenz ratio, $L/L_0\equiv {\kappa }_e/L_0\sigma T\equiv \rho (T)/w(T)$, vs $T$. Dashed lines show the ratio of the low-temperature power laws, namely $({\rho }_0+a{T}^{\alpha })/(w_0+b{T}^{\alpha })$, with $\alpha =3/2$ and 2, for 5.25 and 10 T, respectively. The quasiparticle temperature, $T_{\mathrm{QP}}$, marked by arrows, is defined as the temperature below which $L(T)$ starts to rise, aiming towards unity.

Figure 3

Evolution of characteristic energy scales in ${\mathrm{CeCoIn}}_5$ vs magnetic field. The Fermi-liquid temperature $T_{\mathrm{FL}}$ is the end of the $T^2$ regime in $w(T)$ (squares). The quasiparticle temperature $T_{\mathrm{QP}}$ is the onset of the low-$T$ upturn in $L(T)$ (diamonds). The spin-fluctuation temperature $T_{\mathrm{SF}}$ is reached when $\delta (T)=0$ at high $T$ (circles). Error bars for $T_{\mathrm{QP}}$ and $T_{\mathrm{FL}}$ are smaller than the size of symbols. Note that $T_{\mathrm{QP}}=T_{\mathrm{FL}}$ at $H=10\mathrm{T}$ and above. To the right, we also show $T_{\mathrm{SF}}$ and $T_{\mathrm{FL}}$ for ${\mathrm{CeRhIn}}_5$ (at $H=0$).

Figure 4

Difference between electronic thermal resistivity and electrical resistivity of ${\mathrm{CeCoIn}}_5$, labeled $\delta$, as a function of reduced temperature $T/T_{\mathrm{SF}}$, where $T_{\mathrm{SF}}$ is obtained by making all curves at different fields (as indicated) match at high temperature. The corresponding data for antiferromagnetic ${\mathrm{CeRhIn}}_5$ in zero field is also shown (thick black line). Inset: temperature dependence of electrical resistivity at the critical field, with the arrow indicating the position of $T_{\mathrm{SF}}$. The line is a linear fit to the data above 8 K.