Zero-Field Quantum Critical Point in ${\mathrm{CeCoIn}}_5$

Quantum criticality in the normal and superconducting states of the heavy-fermion metal ${\mathrm{CeCoIn}}_5$ is studied by measurements of the magnetic Grüneisen ratio ${\Gamma }_H$ and specific heat in different field orientations and temperatures down to 50 mK. A universal temperature over magnetic field scaling of ${\Gamma }_H$ in the normal state indicates a hidden quantum critical point at zero field. Within the superconducting state, the quasiparticle entropy at constant temperature increases upon reducing the field towards zero, providing additional evidence for zero-field quantum criticality.

Figure 1

Magnetic Grüneisen ratio divided by magnetic field ${\Gamma }_H/H$ of ${\mathrm{CeCoIn}}_5$ in the normal conducting state plotted against temperature for fields applied along (a) [001] and (b) [100]. The inset displays ${\Gamma }_{H}H$ as a function of temperature for different field angles close to the respective upper critical fields. Labels 18° and 70° denote field angles from [100] towards [001]. The applied magnetic field is 5, 6, 10, and 12 T for $H\parallel [001]$, 70°, 18°, and parallel [100], respectively.

Figure 2

Magnetic Grüneisen ratio of ${\mathrm{CeCoIn}}_5$ for the field along [001] as ${\Gamma }_{H}H$ versus $T/H^{3/2}$, on double logarithmic scale. The solid grey line indicates a phenomenological fit by the function $f(x)$ (see the main text for the definition). The positions of the maximum and inflection points, respectively, 0.015 and $0.006\mathrm{K}{\mathrm{T}}^{-3/2}$, that define the crossover between the NFL and FL regimes, are indicated by arrows. Inset (a) shows ${\Gamma }_H^{-1}$ within the FL regime for $T\to 0$ versus magnetic field for different field directions. The solid grey line indicates a divergence of the magnetic Grüneisen ratio as ${\Gamma }_H=0.85/H$. Inset (b) shows the weighted mean-square deviation from the phenomenological function $f(x)$ versus the quantum critical field $H_c$.

Figure 3

$T\mathrm{\text{−}}H$ phase diagram of ${\mathrm{CeCoIn}}_5$ for magnetic fields applied along [001]. The color code in the normal conducting state represents ${\Gamma }_H/H$. The onset of FL behavior determined by thermal expansion [11] (solid circles) and Hall effect [19] (open circles) measurements is included. The two broken lines represent the crossover between the NFL and FL regions $T_{\mathrm{FL}}=0.015\mathrm{K}{\mathrm{T}}^{-3/2}{H}^{3/2}$ and $T_{\mathrm{FL}}=0.006\mathrm{K}{\mathrm{T}}^{-3/2}{H}^{3/2}$, as defined by the maximum position and inflection point in ${\Gamma }_{H}H$, respectively, (see arrows in Fig. 2).

Figure 4

Electronic specific heat divided by temperature $C_{\mathrm{el}}/T$ (open black circles, left axis) and magnetic Grüneisen ratio ${\Gamma }_H$ (solid red circles, right axis) of ${\mathrm{CeCoIn}}_5$ as a function of applied field along [001] at 0.1 K. The inset shows increments of isothermal electronic entropy as a function of field at 0.1 K.