Spin Polarons in Flat Band Ferromagnets

Abstract

Spin polarons are bound states of electrons and spin-flips that form above spin polarized electronic insulators.These bound states conventionally form in one of two settings: in frustrated lattices with dispersive bands -- where the motion of an electron preferences binding a nearby spin-flip -- or in topological flat bands -- where the Chern number enforces an effective dipolar interaction between electrons and spin flips. In this work, we report the formation of a spin polaron in a context that doesn't fall cleanly into either of these paradigms. In particular, we study the one-dimensional Mielke-Tasaki chain, a paradigmatic model of flat band ferromagnetism, which has an exact ferromagnetic ground state, trivial band topology, and quenched kinetic energy in its lowest band. Despite these features, our density matrix renormalization group simulations reveal the presence of spin polarons upon electron doping this model. More surprisingly, combining these numerics with analytic calculations, we show that polaron binding occurs when the interaction-induced kinetic energy of the model is zero -- contrary to intuition from kinetic magnetism -- and the glue binding the electrons and spin-flips arises from weak mixing with the model's dispersive band -- contrary to what occurs in topological flat bands. Our results open the doors to exploring how the quantum geometry of flat bands drives the formation of exotic charge carriers.

Figures (9)

• Figure 1: Spin Polaron Formation in a Flat Band Ferromagnet.(a) We explore spin polaron formation in the 1D Mielke-Tasaki chain---a model of repulsively interacting electrons on the sawtooth lattice whose hoppings are set by a dimensionless parameter $\delta$. (b) At all $\delta$, this model has an exact flat band (red) as well as a dispersive band (blue) separated by a gap $t\delta^2$. (c) Also, the ground state at half-filling of the flat band is an $SU(2)$ symmetry-breaking ferromagnet for any repulsive on-site interaction $U>0$. (d) We study the nature of low-energy charge carriers doped above the flat-band ferromagnet by numerically computing the binding energy $\Delta E_{e \sigma}$ [defined in Eq. \ref{['eq:binding']}] between doped electron and spin flip (magnon) excitations of the ferromagnet. In some regimes of $\delta$ and the interaction strength $U$, this binding energy is zero and consequently electrons are the lowest energy charge carriers and remain decoupled from any added spin flips in the system (shown on top). Importantly, in other regimes, the binding energy is negative and consequently spin polarons (shown on bottom) form.
• Figure 2: Numerical Evidence of Polaron Formation. By simulating the Mielke-Tasaki chain using DMRG, we find a window of hopping ratios $\delta$ and a threshhold interaction strength $U$ above which the model hosts a spin polaron bound state. (a) In particular, we first plot the binding energy of the polaron $\Delta E_{e\sigma}$ [see Eq. \ref{['eq:binding']}] as a function of $\delta$ for different values of $U$ above the observed threshold value of $U_c/t \approx 0.43$. We find a negative $\Delta E_{e\sigma}$ in a window of $\delta$, indicating binding of electrons and spin-flips. We note that at the weakest interaction strength, where the physics of the flat band plays the largest role, a polaron first forms around $\delta_c \approx 1.155$ (see inset and vertical dashed line). (b) We summarize our numerical investigations by plotting the negative of the polaron binding energy, shown in blue (gray) at parameters with (without) polaron formation. We provide a theoretical estimate of the threshold $U$ for $\delta < \delta_c$ (dashed curve) based on considerations of the interaction-induced dispersion of the electron, which we explain in detail in the section on polaron energetics.
• Figure 3: Polaron Formation and Interaction-Induced Dispersion. In contrast to intuition from kinetic magnetism Schrieffer_1988haerter2005kineticsposetti2014classicallisandrini2017evolutionZhang_2018Davydova_2023Morera_2024Morera_Navarro_2024morera2024itinerantmagnetismmagneticpolarons, in the Mielke-Tasaki model, polarons form when the interaction-induced dispersion of electrons is minimized. (a) To see this, we first plot the interaction-induced (Hartree) dispersion of doped electrons that arises due to their interaction with the density of the background ferromagnet, $[E_H(k)-E_H(0)]/U$. This dispersion is perfectly flat for $\delta < 2/\sqrt{3} \approx 1.155$, precisely where we see the polaron arise at the weakest interaction strengths. In contrast, polarons form at much larger interaction strength when $\delta \neq \delta_c$ where the dispersion is sizable. (b) To solidify the relationship between polaron formation and a flat Hartree dispersion, we show that one can tune the location $\delta$ where the dispersion vanishes using a staggered interaction [discussion below Eq. \ref{['eq:Hartreedensity']}]. By plotting the polaron binding energy as a function of $\delta$ for different staggering ratios $U_A/U_B$ and a weak average interaction $(U_A + U_B)/2 \approx \{0.45,0.60,0.75\}$ (lightest to darkest), we find that the onset of polaron formation tracks the location that the dispersion vanishes $\delta \approx 1.155, 1.05,0.95,0.75$ (vertical dashed lines). (c) Furthermore, when $\delta < \delta_c$, we find that the second-order (Schrieffer–Wolff) correction to the single electron dispersion $E_{\mathrm{SW}}(k)$ can approximately cancel the Hartree dispersion at a sufficient interaction strength due to its peaked shape. This provides a predictive heuristic for the onset of polaron formation in $U$ for $\delta < \delta_c$, shown in Fig. \ref{['fig:numerical-evidence']}(b).
• Figure 4: The Mielke-Tasaki Chain.
• Figure 5: The presence of spin polarons was investigated in the parameter range $0.2\le \delta \le 2$ and $0 < U/t \le 50$. For $U/t\gtrsim 5$, the polaron binding energies decrease and the range of $\delta$ supporting the polaron also gets narrower while shifting to larger $\delta$. Polaron formation is observed to cease at $U/t \gtrsim 50$.