Cluster algorithms for quantum impurity models and mesoscopic Kondo physics

Nanoscale physics and dynamical mean-field theory have both generated increased interest in complex quantum impurity problems and so have focused attention on the need for flexible quantum impurity solvers. Here we demonstrate that the mapping of single-quantum impurity problems onto spin chains can be exploited to yield a powerful and extremely flexible impurity solver. We implement this cluster algorithm explicitly for the Anderson and Kondo Hamiltonians, and illustrate its use in the “mesoscopic Kondo problem.” To study universal Kondo physics, a large ratio between the effective bandwidth $D_{\mathrm{eff}}$ and the temperature $T$ is required; our cluster algorithm treats the mesoscopic fluctuations exactly while being able to approach the large $D_{\mathrm{eff}}∕T$ limit with ease. We emphasize that the flexibility of our method allows it to tackle a wide variety of quantum impurity problems; thus, it may also be relevant to the dynamical mean-field theory of lattice problems.

Figure 1

Comparison of the local susceptibilities $\chi$ obtained using the spin-chain algorithm (circles) and the Hirsch and Fye algorithm (open squares) in the symmetric Anderson model for $N=5000$, $2D=40$, and $U∕\Gamma =8∕1.6(T_K=0.146)$. $\chi T_K$ is in units of ${(g{\mu }_B∕2)}^2$. Three different realizations are shown: (a) Clean case and (b) two realizations drawn from GOE. (c) The real and imaginary parts of the thermal Green’s function vs Matsubara frequencies for the clean case at $T∕T_K=0.55$. (d) The time required in the spin-chain algorithm to obtain two percent errors on $\chi$ as a function $T_K∕T$. In the low-temperature regime, an almost quadratic dependence in $1∕T$ is observed [compared to $1∕T^3$ for a single sweep at fixed $\Delta \tau$ in the Hirsch and Fye algorithm (Ref. 6)]. The value of $T_K$ used was obtained from the two-loop RG equation of the Kondo model.

Figure 2

(Color online) Local susceptibilities in the symmetric Anderson model for various values of $U$ at fixed $8\Gamma ∕\pi U=J\rho =1∕\pi ,N=1000,2D=10(T_K=0.122)$ for a flatband. Data for the Kondo model at the same $J$ and $\rho$ is also shown. The value of $T_K$ is obtained from the two-loop RG equation in the Kondo model.

Figure 3

(Color online) Local susceptibilities using logarithmic blocking, for the three realizations shown in Fig. 1—the clean case (top) and with mesoscopic fluctuations (bottom). The values of $N_{\Lambda }$ are roughly 80 at $\Lambda =1.2$, 40 at $\Lambda =1.4$, and 24 at $\Lambda =2$. The Kondo temperature $T_K\left(\Lambda \right)$ is obtained using the renormalized $J_{\mathrm{eff}}\left(\Lambda \right)$ for logarithmic discretization (Ref. 26): $T_K=0.145$, 0.141, and 0.132, for $\Lambda =1.2$, 1.5, and 2.0, respectively.