Crossover from self-trapped bound states to perturbative scattering in the Heisenberg-Kondo lattice model

TL;DR

We address transport in a two-dimensional ferromagnetic Heisenberg-Kondo lattice by coupling tight-binding electrons to classical spins and generating spin backgrounds with Langevin dynamics, then computing conductivity from exact eigenstates via the Kubo formula. The study maps a transport phase diagram in the $n$–$J'/t$ plane, identifying a polaronic window at low density and strong coupling near $T_c$ and a perturbative scattering regime elsewhere, uncovering a near-universal metallic resistivity form $\rho(T,n,J') = \rho_{\infty}(n,J')\,f(T/T_c)$ that collapses data across parameters. In the polaronic regime, a nonmonotonic $\rho(T)$ with a peak near $T_c$ emerges, tied to partial localisation and a mobility edge approaching the chemical potential; the resulting excess resistivity scales with $\rho_{\infty}$ and stems from enhanced scattering rather than complete localization. The work links polaron formation, mobility-edge physics, and transport in a strongly coupled electron-spin system, with implications that extend beyond two dimensions.

Abstract

We map out the complete transport phase diagram of the ferromagnetic Heisenberg-Kondo lattice model in two dimensions. The model involves tight-binding electrons with hopping $t$, coupled to classical spins with coupling $J'$, while the spins have a nearest neighbour coupling $J$ between them. We work with a fixed, small $J/t$, and study the temperature dependence of resistivity for varying electron density $n$ and coupling $J'/t$. Our magnetic configurations are generated by exact diagonalisation-based Langevin dynamics, while the conductivity is computed using the Kubo formula on exact eigenstates. We work on lattices of size $20 \times 20$ and can access electron density down to $n \sim 0.01$. The electron system remains homogeneous either when the mean density is large or when the coupling $J'$ is small. In these situations, the resistivity $ρ(T)$ displays a monotonic increase with temperature and can be understood within a perturbative framework. However, at very low density $n \lesssim 0.05$, strong coupling $J'/t \gtrsim 1$, and for $T \sim T_c$, the electrons can locally polarise the magnetic state, create a trapping potential, and form a bound state in it. The resistivity associated with this polaronic phase is distinctly non-monotonic, with a peak near $T_c$. We establish the boundary that separates the many-body polaronic window from traditional scattering and extract a universal form for the resistivity in the scattering regime. We suggest the origin of the `excess resistivity' in the polaronic regime in terms of an increasing fraction of localised states as the temperature tends to $T_c$. This pushes the mobility edge towards the chemical potential $μ$ and results in enhanced scattering of momentum states near $k_F$. While our specific results are in two dimensions, the phenomenology we uncover should be valid even in three dimensions.

Figures (12)

• Figure 1: Transport phase diagram. (a) The polaronic and non polaronic windows in the $n-J'$ plane. In the non polaronic window, the resistivity $\rho(T)$ has a monotonic rise through $T_c$ to a saturated high $T$ behaviour. In the polaronic regime, there is a peak in the resistivity around $T_c$. This peak feature correlates with enhanced localisation of electronic states around $T_c$. (b) The temperature window over which polaronic effects are visible, identified through the resistivity.
• Figure 2: Temperature dependence of magnetisation for different $n$ and $J'$. (a)-(b) Variation of $J'$ at low density $n=1.5\%$, and somewhat larger density, $n=5\%$, respectively. (c)-(d) Variation of $n$ at $J'/t=0.5$ and $2.0$. The highest density we have studied in this paper is $n \sim 12\%$, that data is not shown here to keep the $T$ scale manageable.
• Figure 3: Scaling of the magnetisation, and $T_c$ scales. (a) Plot of $m(T/T_c)$ to check that the overall behaviour has a similar pattern independent of $n$ and $J'$. The approximate collapse will allow us to encode magnetic information only in terms of $T_c$. The legend indicates $(n, J'/t)$ values corresponding to each curve. (b) $T_c(J')$ for two values of $n$. (c) $T_c(n)$ for two values of $J'$.
• Figure 4: Resistivity $\rho(T)$. (a) Density variation at weak coupling, $J'=0.5t$. Note the monotonic $T$ dependence of $\rho(T)$ and the increasing high $T$ saturation value as $n$ is lowered. (b) 'High density' $n=12.5\%$ case for varying $J'$. The temperature dependence shares a similarity with (a). (c) Increasing $J'$ at low density $n=1.5\%$. While the weak coupling resistivity is monotonic, for $J' \ge t$ the resistivity is distinctly non monotonic, with a peak near $T_c$. The high $T$ value also increases with $J'$. Essentially, at low $n$ one enters an unusual transport regime with increasing $J'$. Finally, (d) Increasing density, starting from low $n$, in the strong coupling regime. Here, the peak structure weakens with increasing $n$ and almost vanishes for $n = 5\%$.
• Figure 5: Fitting functions for the resistivity $\rho(T)$. (a) The $J'$ dependence fitted to a form $J'^2/(1 + \alpha J'^2)$ shows a reasonable fit with $\alpha = 1.72$ for all $n$. (b) A fit to the $n$ dependent prefactor to a form $A/n$ in the window $0-10\%$, gives $A \sim 1.2$. (c) The scaled resistivity $\rho(T/T_c)/\rho_{\infty}$ for some combinations of $n$ and $J'$. (d) The best fit $f(T/T_c)$ to the normalised $T$ dependence shown in (c).