Natural orbitals renormalization group approach to the two-impurity Kondo critical point

The problem of two magnetic impurities in a normal metal exposes the two opposite tendencies in the formation of a singlet ground state, driven respectively by the single-ion Kondo effect with conduction electrons to screen impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction between the two impurities to directly form impurity spin singlet. However, whether the competition between these two tendencies can lead to a quantum critical point has been debated over more than two decades. Here, we study this problem by applying the newly proposed natural orbitals renormalization group method to a lattice version of the two-impurity Kondo model with a direct exchange $K$ between the two impurity spins. The method allows for unbiased access to the ground state wave functions and low-lying excitations for sufficiently large system sizes. We demonstrate the existence of a quantum critical point, characterized by the power-law divergence of impurity staggered susceptibility with critical exponent $\gamma =0.60\left(1\right)$, on the antiferromagnetic side of $K$ when the interimpurity distance $R$ is even lattice spacing, while a crossover behavior is recovered when $R$ is odd lattice spacing. These results have ultimately resolved the long-standing discrepancy between the numerical renormalization group and quantum Monte Carlo studies, confirming a link of this two-impurity Kondo critical point to a hidden particle-hole symmetry predicted by the local Fermi liquid theory.

Figure 1

Schematic view of the phase shift ${\delta }_{e\left(o\right)}$ in the even (odd) channel of the conduction band at the Fermi energy.

Figure 2

Schematic view of a standard two-impurity Kondo model with impurity separation $R=3$.

Figure 3

Finite size extrapolation of impurity staggered susceptibility ${\chi }_s$ at $K=0$ and at $J=0.728$ and $0.8$ respectively for $R=1$. Here $L=22$, 24, 30, $\cdots$, 510, and $512,$ respectively. The endpoint at $1/L=0$ of each curve is determined by a quadratic polynomial fit of the leftmost four points of that curve calculated by NORG. Here the NORG results are very consistent with those of Fye's QMC calculations [16].

Figure 4

Impurity staggered susceptibility ${\chi }_s$ and interimpurity spin-spin correlation $〈{\mathbf{S}}_1·{\mathbf{S}}_2〉$ (inset) as functions of $K$ for $J=1$ and $R=2$ with different lengths $L$. It is similar to the case of $J=1$ and $R=4$ (not shown). For $L=4n+2$ represented by solid lines [38], ${\chi }_s$ shows a divergent peak at $K=K_c^{\mathrm{odd}}\left(L\right)$ which moves to the right and finally converges to $K_c\equiv K_c(L=\infty )=0.1555$ as $L\to \infty$ [31]. For $L=4n$ represented by dotted lines, ${\chi }_s$ shows a peak (with discontinuity) at $K=K_c^{\mathrm{even}}\left(L\right)$ which moves to the left and finally converges to the same $K_c$ (determined in the $L=4n+2$ case) as $L\to \infty$ and meanwhile the peak height increases and finally diverges. $〈{\mathbf{S}}_1·{\mathbf{S}}_2〉$ jumps where ${\chi }_s$ peaks. The discontinu-ity of $〈{\mathbf{S}}_1·{\mathbf{S}}_2〉$ indicates a quantum phase transition, which is further characterized as a quantum critical point by the divergence of ${\chi }_s$.

Figure 5

Phase diagram of direct exchange interaction $K$ versus Kondo coupling $J$: calculated critical point $K_c$ as a function of $J$. This defines a phase boundary separating the Kondo singlet phase (small $K$) from the interimpurity singlet phase (large $K$) at zero temperature. $K_c\left(J\right)$ is quadratic at the small $J$ limit (lower inset) and linear at the large $J$ limit (upper inset).

Figure 6

Finite-size data collapse for the case of $J=1$ and $R=2$ with different lengths $L$. This gives rise to ${\chi }_s\sim {|K-K_c|}^{-\gamma }$ with $\gamma =0.60\pm 0.01$. $L_0=-9.2$ and $\nu =1$ are determined by fitting data with formula $|K_c\left(L\right)-K_c|\sim {(L-L_0)}^{-1/\nu }$ [31].

Figure 7

Impurity staggered susceptibility ${\chi }_s$ and interimpurity spin-spin correlation $〈{\mathbf{S}}_1·{\mathbf{S}}_2〉$ (inset) as functions of $K$ for $R=1$. $J=1$ is fixed with different lengths $L$. For $L=4n+2, {\chi }_s$ looks diverging at some value of $K=K_c$. But the diverging tendency is suppressed and $K_c$ moves to $-\infty$ as $L\to \infty$ [31, 38]. Therefore, the ${\chi }_s$ curves for $L=4n+2$ and $L=4n$ both converge to the same one in the thermodynamic limit $n\to \infty$. The case of $R=3$ is similar (not shown).

Figure 8

$〈{\mathbf{S}}_1·{\mathbf{S}}_2〉$ for $R=1, J=0.728$, and $K=0$ at zero temperature. $L=22,24,30,\cdots ,1022$. The endpoint at $1/L=0$ of each curve is determined by a quadratic polynomial fit of the leftmost four points of that curve calculated by the NORG. Our NORG result is very consistent with that of Fye's QMC calculation [16].

Figure 9

$K_c/T_K$ versus $1/J$. In the NRG studies [12, 13], $K_c/T_K$ was predicted to be a constant, about 2.2.

Figure 10

The value of $K=K_c\left(L\right)$ at which ${\chi }_s$ diverges for $R=2, J=1$, and $L=4n+2$. The dots are calculated by NORG. The line is determined by fitting the dots with formula $K_c\left(L\right)=K_c-a{(L-L_0)}^{-1}$, where $K_c, a$, and $L_0$ are fitting parameters. This perfect fitting shows that $|K_c\left(L\right)-K_c|\sim {(L-L_0)}^{-1/\nu }$ with $\nu =1$.

Figure 11

Finite size extrapolation of $K_c$ for $R=4$ and various values of $J$. $L=22$, 30, 42, 62, and 126. The case for $R=2$ is similar (not shown). The endpoint at $1/L=0$ of each curve is determined by fitting the NORG data with formula $K_c\left(L\right)=K_c-a{(L-L_0)}^{-1}$, where $K_c, a$, and $L_0$ are fitting parameters. The fitting quality is very good. Figure 10 is a typical example.

Figure 12

The value of $K=K_c$ at which ${\chi }_s$ diverges for $R=3$ and $L=4n+2$. Dots denote the results calculated by the NORG. Lines are determined by fitting the dots with the formula $K_c=-a{(L-c)}^b$, where $a, b$, and $c$ are fitting parameters. The $K_c$ behavior for $R=1$ is similar (not shown). This figure clearly shows that $K_c$ decreases to $-\infty$ as $L\to \infty$.