Impurities near an antiferromagnetic-singlet quantum critical point

Heavy-fermion systems and other strongly correlated electron materials often exhibit a competition between antiferromagnetic (AF) and singlet ground states. Using exact quantum Monte Carlo simulations, we examine the effect of impurities in the vicinity of such an AF-singlet quantum critical point (QCP), through an appropriately defined “impurity susceptibility” ${\chi }_{\mathrm{imp}}$. Our key finding is a connection within a single calculational framework between AF domains induced on the singlet side of the transition and the behavior of the nuclear magnetic resonance (NMR) relaxation rate $1/T_1$. We show that local NMR measurements provide a diagnostic for the location of the QCP, which agrees remarkably well with the vanishing of the AF order parameter and large values of ${\chi }_{\mathrm{imp}}$.

Figure 1

The square of the staggered magnetization $\langle m^2\rangle$ as a function of the impurity concentration $p$ for different $g$'s. In the AF phase with $g=2<g_c(p=0)=2.522$, impurities reduce the order. The effect of impurities near the QCP and in the singlet phase is discussed in the text. The inset: Finite-size scaling of $\langle m^2\rangle$ for $p=0.01$. The position of the QCP is increased to $g_c(p=0.01)=2.65$. Data were averaged over 120 disorder realizations. The inverse temperature is $\beta =80$.

Figure 2

The impurity susceptibility ${\chi }_{\mathrm{imp}}$ is sharply peaked at $g_c$ (vertical dashed line): Impurities induce AF order. Away from $g_c,{\chi }_{\mathrm{imp}}<0$: Impurities reduce the AF order parameter. The inset shows the $g$ dependence of $\langle m^2\rangle$ for $p=0.01$ (squares) and the clean system (circles); these data were used to obtain ${\chi }_{\mathrm{imp}}$. Both the shift in $g_c$ and the large effect of impurities at the QCP are evident. Data for $\langle m^2\rangle$ result from extrapolations to $L=\infty$. The inverse temperature is $\beta =80$.

Figure 3

(a) Correlation length $\xi$ as a function of $g$. Data are shown around a single impurity (square) and for the clean system (circles). (b) $\xi \sqrt{p_c}$ is roughly constant, consistent with a picture where AF order occurs when the mean impurity separation $\langle l\rangle$ is proportional to $\xi$. Data for inverse temperatures $\beta =40,80$ were compared to ensure convergence to the ground state. $L$ up to 100 was used to calculate $\xi$. (c) The AF order parameter at fixed $\beta =80$ exhibits a sharp crossover indicating the position of the enhanced range of AF order created by the spin-1/2 impurities.

Figure 4

(a) $1/T_1$ as a function of $g$ for different values of $\beta$. The linear size is $L=50,60$. The inset: blowup of the crossing point. $g<g_c,g=g_c$, and $g>g_c$. (b) $g$ dependence of $1/T_1$ for a system with a single removed spin. $x=0,1,2,L/2$ are different horizontal distances from the impurity. See the text. For the impurity system the linear lattice size is $L=20,30$.

Figure 5

$T$ dependence of $1/T_1$ for separations $x$ from the impurity. See the text. The inset: determination of the ${\eta }^{\prime }$ exponent. Linear lattices sizes were $L=20,30$, somewhat smaller than in previous figures because of the necessity to compute the imaginary time-dependent correlation functions.

Figure 6

AF correlation $C\left(\mathbf{r}\right)$ of the spin at an impurity site with the other spins in the layer $\alpha =1$. $g=4,3,2.7$, and 2.52. See the text.