Conserved quantities of $SU\left(2\right)$-invariant interactions for correlated fermions and the advantages for quantum Monte Carlo simulations

In the context of realistic calculations for strongly correlated materials with $d$ or $f$ electrons the efficient computation of multi-orbital models is of paramount importance. Here we introduce a set of invariants for the $SU\left(2\right)$-symmetric Kanamori Hamiltonian, which allows us to massively speed up the calculation of the fermionic trace in hybridization-expansion continuous-time quantum Monte Carlo algorithms. We show that by exploiting this set of good quantum numbers the study of the orbital-selective Mott transition in systems with up to seven correlated orbitals becomes feasible.

Figure 1

Table: Maximum and mean block sizes of the Kanamori Hamiltonian using the total number of electrons in the system $N$, the total spin momentum in $z$ direction $S_z$ (second column), and additionally PS (third column) as good quantum numbers for various $N_{\mathrm{orb}}$. Figure: Ratio of CT-HYB runtime (speed-up) we obtain for an equal amount of Monte Carlo steps with and without PS. Three independent measurement of the runtime were done for each point; the average is plotted and the spread of the results is of the order of the symbols. We first consider the lowest-lying multiplet of $H_{\mathrm{loc}}$ as outer state for the trace (good approximation at very low temperatures) and then include progressively more and more of the lowest multiplets. For $N_{\mathrm{orb}}=7$, the lowest four multiplets cover an energy range of about 4.5 eV (i.e., a range of order $U)$.

Figure 2

Spectral weight at the Fermi level $A\left(0\right)$ for the model sketched in the inset with one symmetric band (blue) and the remaining ones symmetrically shifted above and below by $\Delta =0.7D$. This model as been solved for $N_{\mathrm{orb}}=3$ in Ref. 16. We clearly observe an orbital selective region, with the central band Mott insulating (zero spectral weight) and the shifted bands still metallic (nonzero spectral weight) in a finite $U$ interval up to $N_{\mathrm{orb}}=7$.