Krylov implementation of the hybridization expansion impurity solver and application to 5-orbital models

We present an implementation of the hybridization expansion impurity solver which employs sparse-matrix exact-diagonalization techniques to compute the time evolution of the local Hamiltonian. This method avoids computationally expensive matrix-matrix multiplications and becomes advantageous over the conventional implementation for models with five or more orbitals. In particular, this method will allow the systematic investigation of 7-orbital systems (lanthanide and actinide compounds) within single-site dynamical mean-field theory. We illustrate the power and usefulness of our approach with dynamical mean-field results for a 5-orbital model which captures some aspects of the physics of the iron-based superconductors.

Figure 1

(Color online) Comparison between the Green’s functions of a 3-orbital model computed with the matrix method (open symbols, no truncation of the trace) and the Krylov method (full symbols, truncation of the trace to the lowest energy states) for different temperatures (top panel, $U=6$) and interaction strengths (bottom panel, $\beta =50$). The results become indistinguishable at temperatures which are $\lesssim 1\%$ of the bandwidth.

Figure 2

(Color online) Efficiency (number of updates per second) of the different implementations as a function of system size. The models are $n$-orbital impurity models with rotationally invariant Hund coupling at half filling $\left(\mu ={\mu }_{\text{half}}\right)$, $U=6$, $J/U=1/6$, and $\beta =50$. For the matrix method we show results without truncation (diamonds) and with truncation of the trace to the quantum-number sector containing the ground state (triangles). In the Krylov calculation, the trace is truncated to the lowest energy states.

Figure 3

(Color online) Efficiency as a function of chemical potential for the 5-orbital model $\left(U=6,J/U=1/6,\beta =50\right)$. The chemical potentials have been chosen such that the ground state of $H_{\text{loc}}$ has 5, 6, 7, 8, 9, and 10 electrons, respectively. The corresponding degeneracy $N_{\text{tr}}$ is plotted next to the Krylov data.

Figure 4

(Color online) Weight of the different $\left(n_{\uparrow },n_{\downarrow }\right)$ quantum-number sectors for $\mu -{\mu }_{\text{half}}=4.98\left(N_{\text{tr}}=6\right)$, $\mu -{\mu }_{\text{half}}=5\left(N_{\text{tr}}=31\right)$, and $\mu -{\mu }_{\text{half}}=5.02\left(N_{\text{tr}}=25\right)$.

Figure 5

(Color online) Phase diagram of the 5-orbital model in the space of chemical potential and interaction strength for $J/U=1/4$ (top panel) and $J/U=1/6$ (bottom panel). The blue lines with stars show the $n=5$ and $n=6$ Mott lobes for the model without crystal-field splitting. The red line with crosses shows the effect of a $2+3$ crystal-field splitting of magnitude $\Delta =0.5$ (two orbitals shifted down) on the Mott lobes. Both Mott lobes are now contained in an orbital selective Mott phase (boundary marked with black circles) in which the three degenerate bands are insulating and half filled while the two degenerate bands are metallic. Error bars are on the order of the symbol size.

Figure 6

(Color online) Expectation value of $S^2$ as a function of $J/U$ for the five orbital model with $n=6$, $U=2$, and $\beta =50$. Circles show the result for degenerate orbitals, diamonds for a crystal-field splitting $\Delta =0.5$ which shifts two orbitals down.