Hund bands in spectra of multiorbital systems

Spectroscopy experiments are routinely used to characterize the behavior of strongly correlated systems. An in-depth understanding of the different spectral features is thus essential. Here, we show that the spectrum of the multiorbital Hubbard model exhibits unique Hund bands that occur at energies given only by the Hund coupling $J_{\mathrm{H}}$, as distinct from the Hubbard satellites following the interaction $U$. We focus on experimentally relevant single-particle and optical spectra that we calculate for a model related to iron chalcogenide ladders. The calculations are performed via the density-matrix renormalization group and Lanczos methods. The generality of the implications is verified by considering a generic multiorbital model within dynamical mean-field theory.

Figure 1

(a) Sketch of the Hund band accompanying the standard Hubbard bands. (b), (c) Orbital- and momentum-resolved spectral function $A_{\gamma }(k,\omega )$ in the two-orbital Hubbard model for $n=2.5$, $U/W=1.3$, $J_{\mathrm{H}}/U=0.25$, $L=48$ sites, and orbitals (b) $\gamma =2$ and (c) $\gamma =1$. The horizontal line marks the chemical potential $\mu$. (d) Orbital-resolved density of states $A_{\gamma }\left(\omega \right)$. Points depict the corresponding effective generalized Kondo-Heisenberg model (gKH); see the text for details. The arrow points at the Hund band in the itinerant orbital. Results obtained with DMRG using broadening $\eta =0.04$.

Figure 2

(a) Interaction $U$ dependence of the itinerant orbital's density of states (DOS) $A_1\left(\omega \right)$ obtained for the gKH model with $L=48$ sites, $J_{\mathrm{H}}/U=0.25$, and $n=1.5$. Results obtained with DMRG using $\eta =0.04$ broadening. The solid lines represent the atomic-limit transitions. Inset depicts the results for half electron filling $n=1$. (b) Atomic excitation spectrum. For clarity, we mark only the hole-like (electron removal) excitations and show only one spin projection. D, T, S, H labels stand for doublon, triplet, singlet, and holon, respectively. (c)–(e) DOS $A_1\left(\omega \right)$ projected on the specific final configurations: (c) parallel spins, (d) antiparallel spins, and (e) on the holon; see the text for details. Results obtained with Lanczos diagonalization of $L=8$ lattice with broadening $\eta =0.05$.

Figure 3

(a) Sketch of the transitions to final configurations that contribute to the Hund bands of the three-orbital Hubbard model (3oH). For clarity, we present only the representative configurations (while the results are summed over several configurations of the same type). The labeling of the arrows follows Fig. 2. (b)–(d) DOS $A_1\left(\omega \right)$ of the itinerant orbital ($\gamma =1$) of 3oH with $J_{\mathrm{H}}/U=0.25$. Left panels depict $A_1\left(\omega \right)$ as function of the interaction $U$, while right panels show detailed spectra with projections for $U/W=2$. (b)–(d) depict results for $n=4.5$, $n=3.5$, and $n=4$, respectively. The arrows on the right panels point at the Hund bands [70], with “...” denoting the peaks observable only on a lattice. The solid lines in the left column mark the $-2J_{\mathrm{H}}$, $-3J_{\mathrm{H}}$, and $-5J_{\mathrm{H}}$ slopes. The unlabeled peaks are Hubbard bands which have a $U$ dependence [21]. Results obtained with DMRG on an $L=8$ lattice with broadening $\eta =0.1$.

Figure 4

(a) $A_{\gamma }\left(\omega \right)$ calculated with the DMFT method for an orbitally degenerate three-orbital Hubbard model with semicircular DOS, integer filling $n=4$, $U/D=3.8$, and $J_{\mathrm{H}}/D=0.20,\cdots ,0.40$. The half bandwidth $D=1$ is used as the energy unit; see [21] for details. Triangles mark $\omega =-2J_{\mathrm{H}}$ with a constant shift of $-0.1$ [70]. (b) Optical conductivity $\omega \sigma \left(\omega \right)$ vs the interaction $U$. The lines mark the atomic-limit energy scales. Notice the $2J_{\mathrm{H}}$ peak appearing for $U/W>0.8$. Inset: Data for $U/W=1.5$. Results obtained via DMRG for the gKH model with $n=1.5$, $J_{\mathrm{H}}/U=0.25$, $L=24$ sites, and broadening $\eta =0.1$.