Figure 1

(a) Symbols: Molar susceptibility ${\chi }_{c,5f}^{\mathrm{mol}}$ of the $5f$ electrons in ${\mathrm{Th}}_{1-x}{\mathrm{U}}_x{\mathrm{Ru}}_2{\mathrm{Si}}_2$ at $x=0.03$ vs $T$ in magnetic fields between $H=0$ and 5 T ($H\parallel c$) reproduced from Ref. 12. (a),(b) Curves: The real part of the local, dynamic susceptibility of the 2CK model in the presence of magnetic field at $T=0$ computed with density matrix-NRG at Kondo coupling ${\mathcal{J}}_{2\mathrm{CK}}=0.15D$ (in units of bandwidth) with discretization parameter $\Lambda =2$ and keeping at most 2100 multiplets of $\mathrm{U}(1{)}_{\mathrm{spin}}\times \mathrm{SU}(2{)}_{\mathrm{charge}1}\times \mathrm{SU}(2{)}_{\mathrm{charge}2}$ at each NRG step [20]. (c) The crossover scale $T_H$ [defined as the intersect of the low-$\omega$ and high-$\omega$ asymptotes of $\mathrm{Re}\chi (\omega )$, see plot (b)] shows quadratic and linear $H$ dependence in the 2CK and local moment scaling regimes, respectively; $T_K$, the crossover scale between the local moment and 2CK scaling regimes, is $\approx 1.3\mathrm{K}$. Notice that the experimental data for $H=0\mathrm{T}$ can be fitted equally well with every NRG curve where $H⪅0.2T_K$.Reuse & Permissions

Figure 2

(a) $5f$ electronic specific heat coefficient in units of $R$ ($C_{p,5f}/TR$) of ${\mathrm{Th}}_{1-x}{\mathrm{U}}_x{\mathrm{Ru}}_2{\mathrm{Si}}_2$ at $x=0.07$ vs $T$ for $H$ between 0 and 12 T, reproduced from Ref. 18. (b) 2CK specific heat coefficient for different $H$’s from NRG averaged over 10 $z$ values [25] at ${\mathcal{J}}_{2\mathrm{CK}}=0.1D$ (in units of bandwidth) with $\Lambda =2.5$ and keeping at most 3200 multiplets of $\mathrm{U}(1{)}_{\mathrm{spin}}\times \mathrm{SU}(2{)}_{\mathrm{charge}1}\times \mathrm{SU}(2{)}_{\mathrm{charge}2}$ at each NRG step. The two sets of curves display the same trend, but there is a factor of $\approx 4$ difference in the magnitudes (depending on the precise value of $T_K$). We ascribe this difference to experimental inaccuracy, as we observe a factor of $\approx 2$ difference between the magnitudes of $C_{p,5f}/T$ for the same material published in Refs. 11, 18.Reuse & Permissions