Kondo screening and random-singlet formation in highly disordered systems

Abstract

We propose a minimal model to capture the anomalous low-temperature thermodynamics of doped semiconductors, such as Si:P, across the metal-insulator transition. We consider pairs of local magnetic moments coupled to a highly disordered, non-interacting electronic bath that undergoes a metal-insulator transition with increasing doping. Using a large-$\mathcal{N}$ variational approach, we capture both the inhomogeneous local Fermi-liquid and the insulating random-singlet phase, and find that the local moment susceptibility exhibits a robust power-law behavior, $χ(T) \propto T^{-α}$, with $α$ evolving smoothly with doping before saturating in the metal. Our results highlight the competition between Kondo screening and random-singlet formation as the key ingredient in constructing a complete theory for the low-temperature behavior of strongly disordered interacting systems.

Figures (3)

• Figure 1: Two-site Kondo-Heisenberg problems embedded in a disordered bath with varying impurity concentrations $n$. (a) Fraction of solutions as a function of $n$. The pairs either form a random singlet (RS) or are Kondo screened. (b) Magnetic susceptibility $\chi\left(T\right)$ as a function of $T$, on a log-log scale, for several $n$. The full black line is the Curie law. The dashed lines are power-law fits: $\chi\left(T\right)\propto T^{-\alpha}$. Inset: exponent $\alpha$ as a function of $n$. The vertical dot-dashed line marks the MIT for the conduction bath at $n_c=0.016$. We considered $L=30$.
• Figure 2: (a) Normalized distribution of the decimated exchange couplings $Q\left(J_{ab}\right)$ for three different values of impurity concentration $n$, with $J_0=4t_0^2/U$. Inset: Distribution of the nearest-neighbor couplings $P\left(J_{ab}\right)$ for the same $n$; (b) Magnetic spin susceptibility $\chi\left(T\right)$ as a function of the temperature $T$ for several $n$ on a log-log scale. The dotted lines are fits to a power law, $\chi\left(T\right) \sim T^{-\alpha}$ with the concentration dependence of the exponent $\alpha$ shown in the inset. The black line is the Curie law $\chi\left(T\right) = T^{-1}$. We considered $L=30$.
• Figure 3: Distribution of single-impurity Kondo temperatures $P\left(T_{K}\right)$ for different impurity concentrations $n$ on a log-log scale. The dashed lines are power-law fits of the form $T_{K}^{-\alpha}$. We have $\alpha\approx0.6$ for all $n$, see inset; (b) Susceptibility $\chi\left(T\right)$ as a function of $T$ on a log-log scale. The dashed lines are fits $\chi\left(T\right)\propto T^{-\alpha}$ with the exponent $\alpha \approx 0.6$ displayed in the inset. The full black line is the Curie law. We considered $L=30$. (c) Fraction spins that are not Kondo screened $f(n,L)$, as a function of the inverse system size $L$ and several $n$ on a log-log scale. (d) Extrapolated fraction of free spins in the thermodynamic limit, $f_\infty(n)$, as a function of $n$. Values of $f_\infty(n)$ are obtained from finite-size scaling fits of $f(n,L) = f_\infty(n) + A(n)L^{-\beta(n)}$. For $n \gtrsim 0.012$, the fit returns $f_\infty=0$.