Hybridization expansion impurity solver: General formulation and application to Kondo lattice and two-orbital models

A recently developed continuous time solver based on an expansion in hybridization about an exactly solved local limit is reformulated in a manner appropriate for general classes of quantum impurity models including spin exchange and pair hopping terms. The utility of the approach is demonstrated via applications to the dynamical mean field theory of the Kondo lattice and two-orbital models. The algorithm can handle low temperatures and strong couplings without encountering a sign problem.

Figure 1

(Color online) Every Monte Carlo configuration can be represented by a sequence of operators on the time interval $0⩽\tau <\beta$ (we let time run from right to left to be consistent with the time ordering convention). Different colors correspond to different flavors, while full (empty) circles represent creation (annihilation) operators. The Monte Carlo moves consist of random insertions or deletions of pairs of operators in the different channels.

Figure 2

(Color online) If the creation and annihilation operators for each flavor must occur in alternating order, as is the case for models without exchange and pair hopping, then it is convenient to represent the configurations with nonzero trace by collections of segments. The weight of a segment configuration is determined by the length of the segments and the overlap between segments of different flavors (indicated by the hashed regions).

Figure 3

Distribution of perturbation orders $p\left(k\right)\equiv p\left(k_{\uparrow }\right)=p\left(k_{\downarrow }\right)$ for the ferromagnetic (top panel) and antiferromagnetic (bottom panel) Kondo lattice models at half filling and inverse temperature $\beta t=50$. Note the different $J$ ranges in the two panels. The mean perturbation order shifts lower as the coupling magnitude $\mid J\mid$ is increased. For antiferromagnetic coupling, this effect is much more pronounced.

Figure 4

Top panel: local Green functions for the half filled ferromagnetic Kondo lattice model at the indicated exchange values and temperature. For $J∕t\gtrsim -3$ the computed Green functions are very close to the $J=0$ value and for $J∕t⩾-6$ the long time behavior is characteristic of a metal. The exponential drop of the Green functions for $J∕t⩽-8$ shows that the system becomes insulating at these large couplings. Bottom panel: dependence of the density on chemical potential. The smooth behavior for $J∕t⩾-6$ shows the absence of a gap at half filling whereas a gap is clearly evident in the curve corresponding to $J∕t=-10$. Some suggestion of a precursor to a gap is visible at $J∕t=-6$.

Figure 5

Imaginary time correlation function for the local moment calculated for ferromagnetic (top panel) and antiferromagnetic (bottom panel) couplings at half filling for the $J$ values indicated. Solid lines show results for $\beta t=50$. The dotted lines show results for $\beta t=100$ and $J∕t=-3$ (ferromagnetic case) and $J∕t=0.6$ (antiferromagnetic case).

Figure 6

Top panel: imaginary part of the electron self-energy $\Sigma \left(i{\omega }_n\right)$ for the half-filled ferromagnetic Kondo lattice model at $\beta t=50$, $J∕t=-1$, $-3$, $-6$, and $-10$. The metal-insulator transition which takes place between the last two values of $J$ is obvious from the change in the low-frequency behavior. Bottom panel: expanded view of the low frequency behavior of the self-energy in the smaller-$J$ “metallic” phase. Dashed lines demonstrate an approximate power-law decrease of $\mathrm{Im}\Sigma$ as $\omega \to 0$ with exponents 0.25 for $J∕t=-1$, 0.45 for $J∕t=-3$ and 0.55 for $J∕t=-6$. The solid line is proportional to the theoretically expected (Ref. 21) asymptotic behavior $\mathrm{Im}\Sigma \sim 1∕{\left(\ln {\omega }_n\right)}^2$.

Figure 7

Top panel: Thick lines show the local Green functions for the antiferromagnetic Kondo lattice model for $J∕t=0.4,0.6,0.8,1$ and inverse temperature $\beta t=50$. Thin dash-dotted lines correspond to $\beta t=100$ and $J∕t=0.8,1$. Bottom panel: Density per spin plotted as a function of chemical potential. The data for $J∕t=1.0$ and 1.2 are consistent with the opening of a charge gap.

Figure 8

Top panel: spin gap ${\Delta }_s$ and charge gap ${\Delta }_c$ extracted from fits of the function $a\cosh \left[\Delta (\tau -\beta ∕2)\right]$ to the spin-spin correlation functions and Green functions obtained for $\beta t=100$ and the indicated values of $J$. Bottom panel: same data plotted as a function of $t∕J$ on a semilog scale. The results are consistent with the expected small-$J$ behavior $\ln \Delta \sim -1∕\rho J$.

Figure 9

Ferromagnetically coupled Kondo lattice. Top panel: Green functions obtained for $J∕t=-1$ and $\beta t=10$, 20, 30, 40, and 50. A magnetic transition, setting in at $\beta t\approx 30$ is evident from the appearance of a difference between spin up and spin down Green functions. Bottom panel: staggered magnetization $m=n_{\uparrow }-n_{\downarrow }$ as a function of temperature. Solid lines: $S=1∕2$ model, $J∕t=-1$, and $-2$. Dashed lines: results from the classical model for $J$ corresponding to the same diagonal spin splitting.

Figure 10

Staggered magnetization of the antiferromagnetically coupled Kondo lattice model (half filling, bipartite lattice, particle-hole symmetry). Top panel: staggered magnetization as a function of $J∕t$ for $\beta t=20$, 40, and 80. There is an antiferromagnetic state at small coupling (for sufficiently low temperature) and around $J∕t=1$ a quantum phase transition to a paramagnetic insulator. The dashed line shows the $T=0$ result for classical spins. The bottom panel plots $m=n_{\uparrow }-n_{\downarrow }$ as a function of temperature. We find that the transition temperature is considerably higher than for ferromagnetic coupling and that the magnetization saturates at a smaller value. This smaller magnetization is the result of stronger quantum fluctuations (Kondo divergence) and singlet formation.

Figure 11

Converged Green functions for the two-orbital model with semicircular density of states at half-filling. The ratio of band widths is $t_2∕t_1=2$ and the temperature $\beta t_1=50$. The exchange coupling is fixed as $J=U∕4$. For $U∕t_1=4$ (i.e., $U<U_1^c$), both bands are metallic, whereas for $U∕t_1=8$ (i.e., $U>U_2^c$), both bands are insulating. For $U∕t_1=6$, which lies in between $U_1^c$ and $U_2^c$, the narrow band is insulating, while the wide band is still metallic. Dotted lines show the noninteracting Green functions for $\beta t=50$ and $\beta t=100$.

Figure 12

Top panel: distribution of the orders $p\left(k_a\right)$, $a=1,2$ for $U∕t_1=4$ and $U∕t_1=6$. The average order is lower for the narrow band and decreases with increasing interaction strength. Bottom panel: imaginary part of the self-energies for the metallic states in the narrow and wide band, showing the strong correlations in the narrow band for $U∕t_1=4$ and in the wide band for $U∕t_1=6$. At $U∕t_1=4$, the wide band is only weakly correlated.

Figure 13

Illustration of the relationship between an orbital selective state and the ferromagnetic Kondo lattice model. The top figure shows the spin-spin correlation function in the narrow (insulating) band. The bottom figure shows the imaginary part of the self-energy for the wide (metallic) band. As in the ferromagnetic Kondo lattice model, the correlation function in the insulating orbital saturates at large times, while the self-energy of the metallic band drops very rapidly as ${\omega }_n\to 0$.