Doping $S=1$ antiferromagnet in one-dimension

TL;DR

This work analyzes hole-doped one-dimensional $S=1$ antiferromagnets described by a two-orbital Hubbard-Kanamori chain, focusing on how static and dynamical spin correlations evolve with doping and interactions. Using DMRG and dynamical-DMRG, the authors map out an orbital-RVB region ($x\lesssim 0.3$, $U\sim W$) where the spin sector remains gapped with a coherent magnon peak at $q\approx 2k_F$, alongside a broad high-energy incoherent weight for $q>2k_F$. As $U$ increases toward the ferro-magnetic transition, the system traverses into a spiral-like phase with spin excitations akin to a $J_1$–$J_2$ Heisenberg model; Hund’s coupling further tunes the energy scale through $J=2t^2/(U+J_H)$. The results reveal a coexistence of low-energy coherent magnons and high-energy incoherent spin excitations in the orbital-RVB regime, with clear relevance to doped multiorbital materials and quantum simulators. These insights advance understanding of how itinerant carriers interact with higher-spin backgrounds in one dimension and bear on broader Hund-metal phenomenology in higher dimensions.

Abstract

Antiferromagnetic ground states, when doped, give rise to rich and complex phenomena, prompting detailed investigations in various spin systems. Here, we study the effect of doping on the one-dimensional $S = 1$ antiferromagnetic Heisenberg model (AFM). Specifically, we investigate how the presence of holes affects the static and dynamic (frequency-dependent) spin-spin correlations of the two-orbital Hubbard-Kanamori chain. The latter, at half-filling and in the strong-interaction limit, maps onto an $S = 1$ Heisenberg model. For moderate interactions, an orbital resonating-valence-bond (orbital-RVB) state emerges up to doping levels of $x \lesssim 0.3$. A detailed analysis of interaction strength $U$ and doping concentration $x$ reveals that this phase inherits the key features of spin excitations found in the half-filled case -- namely, a gapped spin spectrum and ``coherent'' magnon behavior up to a wavevector $q$ determined by the Fermi vector, $2k_\mathrm{F} = π(1 - x)$. Furthermore, our results uncover an additional broad, incoherent spectral weight for $q \gtrsim 2k_\mathrm{F}$ at high frequencies. Finally, we show that near the transition to a ferromagnetic phase, a previously unidentified spiral-like state emerges, characterized by spin excitations reminiscent of the $J_1$-$J_2$ Heisenberg model.

Figures (8)

• Figure 1: Sketch of the interaction $U$-hole doping $x$ phase diagram of the two-orbital Hubbard-Kanamori model with degenerated bands. The left $y$-axis represents the interaction strength $U/W$, while the right $y$-axis represents the effective interaction strength $J=2t^2/(U+J_\mathrm{H})$ ($W=4t=2$). Note the log-scale. At $x=0$ and $U>U_\mathrm{c}$ the $S=1$ AKLT-like state is stabilized Jazdzewska2023 (equivalent to $S=1$ Heisenberg model in the $U\,,J_\mathrm{H}\gg t$ limit). The orbital-RVB state with nontrivial topological properties can be found for $0<x\lesssim 0.3$ and $U\sim 2W$Mierzejewski2024. The orbital-RVB region with possible spiral phase is estimated based on the results presented in Sec. \ref{['sec:magstat']}. For $x\ne0$ and $U,J_\mathrm{H}\gg t$ ferromagnetic order is stabelized Moreo2025. A short-range paramagnet (similar to the doped single-band Hubbard model at small $U$) can be found for $U\lesssim W$. White points depict the $(U,x)$ value at which the dynamical spin structure factor $S(q,\omega)$ is evaluated in Sec. \ref{['sec:magdyn']}.
• Figure 2: Interaction $U/W$ dependence of the atomic-level states, i.e., average number of (a) site singlons $n_\mathrm{SS}$ (i.e., each orbital occupied by one electron); (b) orbital singlons $n_\mathrm{SH}$ (i.e., with one electron on one orbital and hole on another); (c) site holons $n_\mathrm{HH}$ (i.e., empty sites); and (d) contributions of sites with double occupancies $n_\mathrm{SD}+n_\mathrm{HD}+n_\mathrm{DD}$. See text for details.
• Figure 3: (a) Average magnetic moment squared $\mathbf{S}^2=(1/L)\sum_i \mathbf{S}^2_i$ dependence on the hole doping $x$ and interaction $U$. The limiting ($U\gg t$) case $\mathbf{S}^2=2$ for $x=0.0$ ($\mathbf{S}^2=0.75$ for $x=0.5$) indicate an effective $S=1$ ($S=1/2$) magnetic moment. (b) The same data plotted as a fraction of the maximal possible magnetic moment $\mathbf{S}^2_\mathrm{max}(x)=2\cdot(1-2x)+0.75\cdot 2x$. The results are presented as $1-\mathbf{S}^2/\mathbf{S}^2_\mathrm{max}$ in the color log-scale. White dashed lines represent approximate borders of the region supporting the orbital-RVB state Mierzejewski2024.
• Figure 4: (a,b) Static spin structure factor $S(q)$ evaluated for (a) fixed interaction strength $U/W=2.6$ and various hole dopings $x=0.0,\dots,0.5$; and for (b) fixed doping $x=0.2$ and various interaction strength $U/W=1.0,\dots,5.7$. (c) The position of maximum $q_\mathrm{max}$ of $S(q)$ in the $U$-$x$ plane of parameters. (d) Detailed interaction $U/W$ dependence (note the log scale) of $q_\mathrm{max}$ for $x=0.1,0.2,0.3,0.4,0.5$. Shaded region depict value of $U$ for which the spiral phase ($q_\mathrm{max}<2k_\mathrm{F}$) is stabilized.
• Figure 5: Doping $x$ dependence of the dynamical spin structure factor $S(q,\omega)$ of the two-orbital Hubbard model for $U/W=3$. Panels (a-e) depict data for $x=0.0,0.1,\dots,0.4$, respectively. The white dashed line in panel (a) depicts the magnon dispersion relation of the $S=1$ Heisenberg model White2008. In each plot, $100$ frequency points are shown ($\Delta\omega=\omega_\mathrm{max}/100$, where $\omega_\mathrm{max}$ is the maximal presented frequency). Left $y$-axis represents frequency $\omega$ in unit of hopping $t$, while right $y$-axis represents $\omega$ in units of $J=2t^2/(U+J_\mathrm{H})$.