Impurity-induced antiferromagnetic domains in the periodic Anderson model

A central feature of the periodic Anderson model is the competition between antiferromagnetism, mediated by the Ruderman-Kittel-Kasuya-Yosida interaction at small conduction electron-local electron hybridization $V$, and singlet formation at large $V$. At zero temperature, and in dimension $d>1$, these two phases are separated by a quantum critical point $V_c$. We use quantum Monte Carlo (QMC) simulations to explore the effect of impurities which have a local hybridization $V_{*}<V_c$ in the antiferromagnetic regime which are embedded in a bulk singlet phase with $V>V_c$. We measure the suppression of singlet correlations and the antiferromagnetic correlations which form around the impurity, as well as the size of the resulting domain. Exact diagonalization calculations for linear chains allow us to verify that the qualitative features obtained at intermediate coupling and finite $T$ persist to strong coupling and $T=0$, regimes which are difficult to access with QMC. Our calculations agree qualitatively with NMR measurements in ${\mathrm{CeCoIn}}_{5-x}{\mathrm{Cd}}_x$.

Figure 1

Exact-diagonalization results for the $f$ orbital impurity susceptibility on near-neighbor sites. ${\chi }_{\mathrm{ff}}^{\mathrm{imp}}(r=1)$ probes the response of AF correlations to a local shift in $V\to V_{*}=V-dV$. ${\chi }_{\mathrm{ff}}^{\mathrm{imp}}(r=1)$ is peaked at intermediate values of $fd$ hybridization: At large $V$, deep in the singlet phase, ${\chi }_{\mathrm{ff}}^{\mathrm{imp}}(r=1)$, is small. Similarly, ${\chi }_{\mathrm{ff}}^{\mathrm{imp}}(r=1)$ also declines as $V\to 0$, deep in the AF regime. A maximum occurs at intermediate $V$. Results are given for on-site $f$ electron repulsion $U_{\mathrm{f}}=4,8,12$.

Figure 2

$ff$ spin-correlation function $c_{\mathrm{ff}}\left(r\right)$ for an impurity with $V_{*}=0.5$ and different bulk $V$. As $V$ approaches the singlet-AF crossover, spin correlations at long range develop. The $f$-orbital on-site repulsion $U_{\mathrm{f}}=8$.

Figure 3

Spin correlations in a system with bulk $V=1.2$ for different impurity $V_{*}$. While there is some increase in the near-neighbor $ff$ spin correlation function $c_{\mathrm{ff}}(r=1)$, values at larger separation are quite insensitive to $V_{*}$. Here the $f$-orbital on-site repulsion $U_{\mathrm{f}}=8$.

Figure 4

The effect of $U_f$ on spin correlations for fixed impurity and bulk hybridizations $V_{*}=0.5$ and $V=1.0$. The moment is nearly saturated even for $U_f=4$, so that the main effect of increasing the on-site repulsion is a significant enhancement of the correlations at larger distances.

Figure 5

The $fd$ singlet correlator $c_{\mathrm{fd}}\left(r\right)$ as a function of distance $r$ from the impurity is shown for bulk hybridization $V=1$ and different impurity hybridizations $V_{*}$. Moving the impurity deeper into the AF phase steadily reduces the singlet directly on the defect site $\left(r=0\right)$. However, the spatial extent of this hole does not increase as $V_{*}$ decreases. Here the bulk hybridization $V=1.0$. The temperature $T=t/30$. The $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 6

Spin correlations between local and conduction electrons, $c_{\mathrm{fd}}\left(r\right)$, are shown for different bulk $V$ and fixed impurity $V_{*}=0.7$. All curves exhibit a similar short-range reduction of singlet correlations near the impurity, however, this reduction is largest for intermediate $V$. (See also Fig. 6.) The temperature $T=t/30$. The $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 7

The reduction of the local-conduction spin correlator, $c_{\mathrm{fd}}(r=0)$, from the asymptotic value $c_{\mathrm{fd}}(r\to \infty )$ shown for different bulk hybridization $V$. The temperature $T=t/30$. Results are shown for on-site $f$ electron repulsion $U_{\mathrm{f}}=4,6$. Here, for $U_{\mathrm{f}}=4$, the hole gets deeper as $V$ is reduced from $V=2.0$ and becomes deepest for $V=1.1$, after which it is again reduced. This tendency fits well with the picture that the effect of an impurity is largest near the AF-singlet QCP, e.g., as we show in Fig. 1.

Figure 8

On-site singlet correlator $c_{\mathrm{fd}}\left(r\right)$ as a function of distance $r$ from the defect. Here the bulk hybridization $V$ is varied at fixed $V-V_{*}=0.4$. The maximum in the reduction $\mathrm{\Delta }{c}_{\mathrm{fd}}\left(V\right)$ at intermediate $V$ is absent. The temperature $T=t/30$ and the $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 9

Spin-correlation function $c_{\mathrm{ff}}\left(r\right)$ between two local $\left(f\right)$ orbitals. A defect with impurity hybridization $V_{*}=0.7$ is present in different background materials with varying $V$. There are long-range correlations for $V$ in the AF regime, but the impurity can only induce AF order locally if $V$ is large. The temperature $T=t/30$ and the $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 10

$ff$ electron spin-correlation function $c_{\mathrm{ff}}\left(r\right)$ for bulk hybridization $V=1.2$ and different impurity hybridizations $V_{*}$. As $V_{*}$ decreases the short-range AF order with near neighbor $(1,0)$ becomes deeper. The temperature $T=t/30$. The $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 11

Spin correlations between local electrons, $c_{\mathrm{ff}}\left(r\right)$, for near neighbors $(1,0),(1,1)$, and $(2,0)$. $c_{\mathrm{ff}}\left(r\right)$ grows as $V$ approaches the AF-singlet crossover. The temperature $T=t/30$. The $f$-orbital on-site repulsion $U_{\mathrm{f}}=4$.

Figure 12

Correlation length vs $V$ given for fixed impurity $V_{*}=0.5,0.7$. Results are shown for two values of the on-site $f$ electron repulsion $U_{\mathrm{f}}=4,6$. OZ (MFT) refer to fits to an Ornstein-Zernike (mean-field theory) form. Although the values of $\xi$ are sensitive to this choice, the main qualitative feature of the curves is not. It shows an evolution from a flat $\xi \approx 2$ at large $V$ to a rising $\xi \approx 4–6$ (at $U_{\mathrm{f}}=4)$, indicating larger AF regions. The temperature $T=t/30$.

Figure 13

Near-neighbor $d$-wave pair correlations as a function of impurity hybridization $V_{*}$ for different bulk hybridizations for $\beta t=30$ and $U_f=4$.