Block orbital-selective Mott insulators: A spin excitation analysis

We present a comprehensive study of the spin excitations—as measured by the dynamical spin structure factor $S(q,\omega )$—of the so-called block-magnetic state of low-dimensional orbital-selective Mott insulators. We realize this state via both a multi-orbital Hubbard model and a generalized Kondo-Heisenberg Hamiltonian. Due to various competing energy scales present in the models, the system develops periodic ferromagnetic islands of various shapes and sizes, which are antiferromagnetically coupled. The $2\times 2$ particular case was already found experimentally in the ladder material $\mathrm{Ba}{\mathrm{Fe}}_2{\mathrm{Se}}_3$ that becomes superconducting under pressure. Here we discuss the electronic density as well as Hubbard and Hund coupling dependence of $S(q,\omega )$ using density matrix renormalization group method. Several interesting features were identified: (1) An acoustic (dispersive spin-wave) mode develops. (2) The spin-wave bandwidth establishes a new energy scale that is strongly dependent on the size of the magnetic island and becomes abnormally small for large clusters. (3) Optical (dispersionless spin excitation) modes are present for all block states studied here. In addition, a variety of phenomenological spin Hamiltonians have been investigated but none matches entirely our results that were obtained primarily at intermediate Hubbard $U$ strengths. Our comprehensive analysis provides theoretical guidance and motivation to crystal growers to search for appropriate candidate materials to realize the block states, and to neutron scattering experimentalists to confirm the exotic dynamical magnetic properties unveiled here, with a rich mixture of acoustic and optical features.

Figure 1

(a) Hubbard-Hund interaction ($U\text{−}{J}_{\mathrm{H}}$) phase diagram of the generic multiorbital Hubbard model. At $U\ll W$ (with $W$ the kinetic energy bandwidth), the system is a paramagnetic metal. At $U\gg W$, the system is a Mott insulator. These two phases are separated, at robust Hund interaction and intermediate $U$, by the orbital-selective Mott phase with at least one orbital Mott localized and the other orbitals displaying metallic behavior. The schematic shapes of the density-of-states are also shown. (b) Magnetic phase diagram of the OSMP. At $U<W$, the system is paramagnetic for all fillings. At the two limiting fillings in the plot, i.e., at half-filling and at one electron above the band-insulator, the state is antiferromagnetic with staggered spin. For large enough repulsion $U\gg W$, ferromagnetic (FM) order is observed for all noninteger values of the electronic filling. For $U\sim W$, the system is in the block-magnetic phase. In between the latter and FM, a block-spiral order dominates. Arrows indicate the representative spin order.

Figure 2

Static structure factor $S\left(q\right)$ of the magnetic orders present in the block-OSMP regime. Top panel: sketch of spin alignment with wave vector $q_{\max }=\pi /l$ for $l=1,2,3,4$. Bottom panel: $S\left(q\right)$ of the static spin structure factor for a given $q_{\max }$. The presented data have 0.5 offset (top to bottom) for clarity. Arrows for $q_{\max }=1/3$ and $q_{\max }=1/4$ indicate additional Fourier modes present for block-magnetic order. Data reproduced from Ref. [46].

Figure 3

Single-particle spectral function function $A(q,\omega )$. (a, b) are for the two-orbital Hubbard model and (c) for the generalized Kondo-Heisenberg model at electronic filling $n_{\mathrm{H}}=2.5/2$ and $n_{\mathrm{K}}=3/2$, respectively. In both cases $L=48$ is used. (a) is in the paramagnetic regime $U/W=0.1$, and (b) in the block-OSMP regime $U/W=1.0$. (c) Results for the OSMP effective Hamiltonian (generalized Kondo Heisenberg model) at $U/W=1.0$. The right panels of (a–c) are the corresponding density of states (DOS). (d) Comparison of DOS between the two models. In all calculations we used frequency resolution $\mathrm{\Delta }\omega =0.02\left[\mathrm{eV}\right]$ and broadening $\eta =2\mathrm{\Delta }\omega$.

Figure 4

Comparison of the dynamical spin structure factor $S(q,\omega )$ between (a) the two-orbital Hubbard and (b) the generalized Kondo-Heisenberg models, as calculated for $L=48,J_{\mathrm{H}}/U=0.25,U=W$, and $n_{\mathrm{H}}=2.5/2$ ($n_{\mathrm{K}}=1/2$). (c) Frequency dependence of $S(q,\omega )$ for $q=\pi /2$ and $q=\pi$.

Figure 5

(a–c) Dynamical spin spin structure factor $S(q,\omega )$ in the orbital-selective Mott regime corresponding to the (a) $\pi /2$-block $\uparrow \uparrow \downarrow \downarrow$, (b) $\pi /3$-block $\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow$, and (c) $\pi /4$-block $\uparrow \uparrow \uparrow \uparrow \downarrow \downarrow \downarrow \downarrow$ phases. Results shown are for $L=48$ sites, $U/W\simeq 1$, and $J_{\mathrm{H}}/U=0.25$ using the generalized Kondo-Heisenberg model. White lines are fits to the dispersion relation ${\omega }_{\mathrm{A}}\left(q\right)=\tilde{J}|sin\left(q{n}_{\mathrm{K}}\right)|$ (with $\tilde{J}=0.035,0.011,0.003$ for $n_{\mathrm{K}}=1/2,1/3,1/4$, respectively).

Figure 6

(a–c) Fourier components of the classical spin patterns for the $\pi /l$-block states, with $l=2,3,4$. Lines represent the components of the Fourier transform, while color (gray) arrows represent spins which contribute (do not contribute) to a given mode. Boxes represent the latter within one magnetic unit cell. (d) Fourier transforms of the classical $\pi /l$-block patterns. $\delta$ functions where broaden by a Gaussian kernel for better clarity in the plot.

Figure 7

Electronic filling $n_{\mathrm{K}}$ dependence of the overall energy scale $\tilde{J}$ of the dispersion relation ${\omega }_{\mathrm{A}}\left(q\right)$. Dashed lines represent fits to a phenomenological expression $\tilde{J}=aexp\left(b{n}_{\mathrm{K}}\right)$. Inset is the same data but in a $y$-log scale. The dashed red horizontal line represents the smallest explicit energy scale present in the generalized Kondo-Heisenberg model, namely, the localized spin-exchange $K$.

Figure 8

(a–f) Hubbard $U$ and (g–l) Hund exchange $J_{\mathrm{H}}$ dependence of the dynamical spin structure factor $S(q,\omega )$, calculated for $L=48$ and $n_{\mathrm{K}}=1/2$. Panels (a–f) depict results for $U/W=0.6,\cdots ,1.4$ and $J_{\mathrm{H}}/U=0.25$, while panels (g–l) for $J_{\mathrm{H}}=0.1,\cdots ,0.35$ and $U/W=1$. The white line in panels (f) and (l) indicate the ${\omega }_{\mathrm{O}}\left(q\right)=0.051+0.005|sin\left(2q\right)|$ dispersion.

Figure 9

(a, b) Frequency $\omega$ dependence of the dynamical spin structure factor $S(q,\omega )$ at $q=\pi$ as calculated for $L=48$ sites. In (a) $U/W=0.6,0.7,\cdots ,1.4$ (top to bottom) at fixed $J_{\mathrm{H}}/U=0.25$, while in (b) $U/W=1.0$ is fixed and we vary $J_{\mathrm{H}}/U=0.10,0.15,\cdots 0.35$ (top to bottom). (c) Hund $J_{\mathrm{H}}$ (bottom $x$ axis) and interaction $U/W$ (top $x$ axis) dependence of the position of the maximum in $S(q=\pi ,\omega )$, at fixed $U/W=1$ and $J_{\mathrm{H}}/U=0.25$, respectively. In addition we show data for the model with $J_{\mathrm{H}}=0.25W$ while varying $U/W$. See text for details.

Figure 10

(a) Sketch of the FM-AFM alternating Heisenberg model $H_{\mathrm{alter}}$, (7). (b) Dynamical spin structure factor $S(q,\omega )$ of the $S=1$ 1D alternating Heisenberg model with $L=64$, $J_{\mathrm{FM}}=J_{\mathrm{AFM}}=1$, and $\delta \omega /J_{\mathrm{FM}}={10}^{-2}$.

Figure 11

Dynamical spin structure factor $S(q,\omega )$ calculated for the 1D $J_1\text{−}{J}_N$ model $H_{1N}$, (8), corresponding to (a) $N=2$, (b) $N=3$, and (c) $N=4$ ($J_1=J_N=1$, $L=64$, $\delta \omega /J_1={10}^{-2}$). On top of each panel we present a schematic representation of each $J_1\text{−}{J}_N$ model.

Figure 12

Dynamical spin structure factor $S(q,\omega )$ of the half-filled antiferromagnetic state. (a) Results for the generalized Kondo-Heisenberg at $n_{\mathrm{H}}=1$ and $U/W=1$, using $L=48$ sites. The dashed line is a fit to the sinelike dispersion, namely, ${\omega }_{\mathrm{A}}\left(q\right)=0.2sin\left(q\right)$. (b) $S(q,\omega )$ of the $S=1$ isotropic Heisenberg model with $J=0.07\left[\mathrm{eV}\right]$. (c) Comparison of results between the gKH and $S=1$ Heisenberg models at wave vectors $q=\pi /2$ and $q=\pi$.