Experimental consequences of quantum critical points at high temperatures

We study the $\mathrm{C}{\mathrm{r}}_{1-x}\mathrm{R}{\mathrm{e}}_x$ phase diagram finding that its phase transition temperature towards an antiferromagnetic order $T_N$ follows a quantum ${\left[\left(x_c-x\right)/x_c\right]}^{\psi }$ law, with $\psi =1/2$, from the quantum critical point (QCP) at $x_c=0.25$ up to $T_N\approx 600\mathrm{K}$. We compare this system to others in order to understand why this elemental material is affected by the QCP up to such unusually high temperatures. We determine a general criterion for the crossover, as a function of an external parameter such as concentration, from the region controlled solely by thermal fluctuations to that where quantum effects become observable. The properties of materials with low coherence lengths will thus be altered far away from the QCP.

Figure 1

(a) Synchrotron x-ray diffraction pattern for the $x=0.246$ sample. The Rietveld fit yields a 97 wt% of bcc Cr phase and only a 3 wt% of hexagonal Re, showing that the sample is mainly monophasic. (b) In-lens detector photograph of the surface showing the small crystallite size ($\sim 30–100\mathrm{nm}$). (c) Measurement of composition by electron backscattering detector. The white spots correspond to silver paint speckles. We observe a homogeneous distribution of Cr and Re. For this sample, the energy dispersive x-ray spectroscopy (EDX) analysis on five different spots gave a Re concentration of 0.187 ± 0.007. The black spots correspond to the $\mathrm{Si}{\mathrm{N}}_x$ substrate.

Figure 2

Electrical resistance for samples with different Re content. Left panel: High temperature behavior of the low concentration region (with residual resistivity subtracted). The $x=0.246$ curve was obtained by cooling under magnetic field [see text and Fig. 3]. The Néel temperature is defined by the steepest increment of the curves, as obtained from their derivative. Right panel: Low temperature, high concentration region, showing the superconducting transitions. The superconducting transition temperature is taken at the onset of the transition.

Figure 3

(a) Double logarithmic plot to show the $T^5$ temperature dependence (straight lines). (b) Residual resistivity as a function of concentration. (c) Hall constant as a function of temperature for several concentrations. (d) Evidence for antiferromagnetic domain ordering due to cooling under a magnetic field of 8 T from temperatures much higher than $T_N$ for the sample nearest to the critical concentration.

Figure 4

Lower panel: Phase diagram of $\mathrm{C}{\mathrm{r}}_{1-x}\mathrm{R}{\mathrm{e}}_x$ as a function of Re content. Previous determinations of $T_N$ are from Ref. [22] (mauve circles) compared to our data (red triangles). The previously reported superconducting transition temperatures (Ref. [12]) are shown (light blue squares) together with our data (blue inverted triangles). It is clear from the data for our more homogenous samples that superconductivity does not coexist with antiferromagnetism, precluding a nonconventional superconductor. We also observe that the decrease of $T_N$ follows the power law $718{\left[\left(0.248-x\right)/0.248\right]}^{0.5}$ from $\sim 600\mathrm{K}$ down to the QCP. Upper panel: For illustration purposes, we show the Fermi surfaces of the Cr-Re alloy at $x=0,0.1,0.2$, and 0.3 illustrating the nesting wave vector that changes from incommensurate (0) to commensurate (0.1; 0.2) and then back to incommensurate (0.3).

Figure 5

Left panel: Evolution of the ordering transition temperatures for different materials showing an exponential decrease with pressure of the ordering temperature at low pressures (black lines), followed at high pressures by the ${[(P_c-P)/P_c]}^{0.5}$ law (blue lines). $\beta -\mathrm{N}{\mathrm{a}}_{0.33}{\mathrm{V}}_2{\mathrm{O}}_5$ (light green circles) (Ref. [30]); $\mathrm{NbS}{\mathrm{e}}_3$ (light brown squares) (Ref. [28]); and Cr (green diamonds) (Ref. [9]). Right panel: Materials that show the ${\left[\left({\delta }_c-\delta \right)/{\delta }_c\right]}^{1/2}$ (δ is either pressure or concentration) law in all the measured range. Inset: Log-log plot to obtain the power of the ${\left[\left({\delta }_c-\delta \right)/{\delta }_c\right]}^{\psi }$; the slopes of the blue lines give a value of $\psi$ for 0.50, 0.49, 0.49, and 0.50 for o-$\mathrm{Ta}{\mathrm{S}}_3$ (orange diamonds) (Ref. [32]), 1T-$\mathrm{TiS}{\mathrm{e}}_2$ (red circles) (Ref. [31]), $\mathrm{BaF}{\mathrm{e}}_2\mathrm{A}{\mathrm{s}}_2$ (mauve squares) (Ref. [33]), and $\mathrm{C}{\mathrm{r}}_{1-x}\mathrm{R}{\mathrm{e}}_x$ (light green triangles), respectively. This type of fit yields a very reliable value of the exponent. For the measured zero temperature coherence lengths at ambient pressure or zero concentration ${\xi }_{\mathrm{meas}}$, see the text and Table 1.

Figure 6

Schematic phase diagram showing the behavior of a typical ordering transition as a function of an external parameter $\delta$. The crossover from the BCS exponential to the quantum power law regime is governed by a crossover ratio $\kappa$ (unity in the figure) of the zero temperature intrinsic correlation length ${\xi }_{\mathrm{coh}}\left(0,\delta \right)$ to the quantum correlation length ${\xi }_q\left(0,\delta \right)$.