Multiorbital Quantum Impurity Solver for General Interactions and Hybridizations

We present a numerically exact inchworm Monte Carlo method for equilibrium multiorbital quantum impurity problems with general interactions and hybridizations. We show that the method, originally developed to overcome the dynamical sign problem in certain real-time propagation problems, can also overcome the sign problem as a function of temperature for equilibrium quantum impurity models. This is shown in several cases where the current method of choice, the continuous-time hybridization expansion, fails due to the sign problem. Our method therefore enables simulations of impurity problems as they appear in embedding theories without further approximations, such as the truncation of the hybridization or interaction structure or a discretization of the impurity bath with a set of discrete energy levels, and eliminates a crucial bottleneck in the simulation of ab initio embedding problems.

Figure 1

Imaginary-time Green’s function for the spinless Anderson model at temperatures $T=t/4$ (top panel), $T=t/64$ (bottom panel), and for intermediate temperatures indicated. Diagonal (left panels) and off-diagonal (right panels) elements for a small discrete bath (see text) obtained with exact diagonalization (dashed black), ct-hyb (solid red) and inchworm method (solid green). ct-hyb data are available only down to $T=t/16$, and the systematic deviation of the ct-hyb results are indicative of an additional ergodicity problem.

Figure 2

Diagonal imaginary-time Green’s function for the Kanamori model at temperature $T=t/8$ (top panel), $T=t/64$ (bottom panel), and intermediate temperatures $T=t/16$ and $T=t/32$, for a discrete band (left panels) and a semicircular band with $t=1$. Results from ct-hyb (solid red) and inchworm method (solid green), along with exact diagonalization where available (black, left panel only).

Figure 3

Scaling analysis. Mean absolute deviation from exact result (top two panels, SAM and discrete Kanamori models) and statistical standard errors (bottom panel, continuous Bethe band Kanamori model) divided by inverse temperature $\beta$, as a function of $\beta$. ct-hyb (red) and inchworm method (green) results are shown along with exponential fits for ct-hyb (light red).