Kondo-Anderson transitions

Dilute magnetic impurities in a disordered Fermi liquid are considered close to the Anderson metal-insulator transition (AMIT). Critical power-law correlations between electron wave functions at different energies in the vicinity of the AMIT result in the formation of pseudogaps of the local density of states. Magnetic impurities can remain unscreened at such sites. We determine the density of the resulting free magnetic moments in the zero-temperature limit. While it is finite on the insulating side of the AMIT, it vanishes at the AMIT, and decays with a power law as function of the distance to the AMIT. Since the fluctuating spins of these free magnetic moments break the time-reversal symmetry of the conduction electrons, we find a shift of the AMIT, and the appearance of a semimetal phase. The distribution function of the Kondo temperature $T_K$ is derived at the AMIT, in the metallic phase, and in the insulator phase. This allows us to find the quantum phase diagram in an external magnetic field $B$ and at finite temperature $T$. We calculate the resulting magnetic susceptibility, the specific heat, and the spin relaxation rate as a function of temperature. We find a phase diagram with finite-temperature transitions among insulator, critical semimetal, and metal phases. These new types of phase transitions are caused by the interplay between Kondo screening and Anderson localization, with the latter being shifted by the appearance of the temperature-dependent spin-flip scattering rate. Accordingly, we name them Kondo-Anderson transitions.

Figure 1

The local intensity is plotted here for a critical state at the three-dimensional AMIT, as obtained by exact diagonalization of the Anderson tight binding model with $W=16.5t$, with hopping parameter $t$ at energy $E=2t$ on a cubic lattice with spacing $a$ and size $L=100a$. The coloring of the plotted intensity is as follows: (red) $\alpha \in [1.2,1.8]$, (green) $\alpha \in [1.8,2.4]$, and (light blue) $\alpha \in [2.4,3.0]$, where is defined by $\alpha =-{\ln \left|\psi \right|}^2/\ln L$. Sites with higher intensity, $\alpha <1.2$, are so rare that their occurrence cannot be resolved in this plot. All other sites whose intensities are not plotted correspond to lower intensity $\alpha >3.0$. Thereby, about 80$\%$ of the total state intensity is shown.

Figure 2

The conditional intensity $I_{\alpha }$ relative to the intensity of an extended state as function of the distance in energy to the mobility edge in units of the correlation energy $E_c$. The exponent $\alpha$, which is related to the intensity $L^{-\alpha }$ at $E_M$, takes the values $\alpha =2,3,3.5,4,6$ in the sequence of decreasing dashing. Local pseudogaps are seen for $\alpha >{\alpha }_0$, while local power-law divergencies occur for $\alpha <{\alpha }_0$. Here we set $d=3$ and ${\alpha }_0=4$.

Figure 3

The distribution of $\alpha$ defined in the metallic regime by Eq. (26), as obtained by exact numerical diagonalization of a 3D sample of size $L^3={128}^3$ in units of the grid cell volume $a^3$. The energy of that state is approximately $E=0$, and a box distribution of uncorrelated disorder potential with $W=15t$ is taken. The analytical expression Eq. (27) is plotted for comparison as the solid line for a correlation length $\xi =22a$ (taken from the numerical results of Ref. 30).

Figure 4

The conditional intensity $I_{\alpha }$ at energy $E_l$, given that a state at the energy $E_k$ has intensity $L^d{\left|{\psi }_k\left(\mathbf{r}\right)\right|}^2={\xi }_k^{d-\alpha }$, relative to the intensity of an extended state. It is plotted as function of the distance of energy $E_l$ from the mobility edge energy $E_M$, in units of the correlation energy $E_c$. The exponent $\alpha$ at the energy $E_k$, takes the value $\alpha =4.5$. The energy $E_k$ is varied from the mobility edge into the metallic regime with $(E_k-E_M)/E_c=0,0.2,0.3,0.4,0.7$, from left to right. Instead of a local pseudogap, one sees an increasingly shallow depression. Here we set $d=3$ and ${\alpha }_0=4$. Note that it varies slowly in a range ${\Delta }_{{\xi }_k}$ around the Fermi energy.

Figure 5

The distribution of Kondo temperatures $T_K$ in units of $T_K^{\left(0\right)}$ in the metallic phase, Eq. (33), is plotted as function of the distance to the mobility edge, $E_F-E_M$, in units of $E_c$ for an exchange coupling $j=1/5$.

Figure 6

The fraction of free magnetic moments $P_{\mathrm{FM}}$ at $T=0$ K, Eq. (37), in a three-dimensional disordered metal as function of the exchange coupling $J$ (in units of the band width $D$) and disorder strength $W$ (in units of the critical value $W_c^O$). Critical correlations result in a finite $P_{\mathrm{FM}}$ even for large $J>J_c^A$ (dashed line). For $J<J^{*}$, Eq. (50) there is a critical region for disorder amplitudes $W_c^O<W<W_c\left(J\right)$, where $W_c(j=J/D)$ is given by Eq. (49). This figure was previously published by the authors in Ref. 21.

Figure 7

The quantum phase diagram in a magnetic field $B$ (arbitrary units), as function of disorder amplitude $W$, in units of $t$. We set $d=3,\eta =2$, and $j=0.2$.

Figure 8

The ratio of free magnetic moments (magnetic moments with $T_K<T$), $n_{\mathrm{FM}}/n_M$, as function of disorder strength $W$ on the insultating side of the transition for temperatures $T/E_c=0$, $1\times {10}^{-6}$, $5\times {10}^{-6}$, $1\times {10}^{-5}$ from the bottom to the top curve. We set $j=0.2$, $d=z=3$, and $\eta /d=2/3$.

Figure 9

The finite-temperature phase diagram of Kondo-Anderson transitions. The solid lines are plots of the critical temperatures $T_c^M(W,J)$, Eq. (68), and $T_c^I(W,J)$, Eq. (66), respectively, where the disorder amplitude $W$ is given in units of the hopping parameter $t$ and the temperature is in ratios of $E_c$. We used the following parameters: $j=0.2$, ${\alpha }_0=4$, $d=3$, $\eta /d=2/3$, $\nu =1.57$, and $a_c^2/\left(D_e{\tau }_s^{\left(0\right)}\right)=0.1$.