Topological Filtering and Emergent Kondo Scale

Abstract

We study the Kondo effect induced by a topological soliton in a one-dimensional Dirac system with the sign-changing mass term. The soliton hosts a localized zero mode whose spatially extended wavefunction leads to a momentum-dependent exchange coupling with itinerant electrons. We show that this structure generates a nontrivial form factor that suppresses high-energy scattering processes, resulting in an energy-dependent effective Kondo coupling. As a consequence, the real-space structure of the soliton directly controls the emergent Kondo scale. This work establishes a mechanism by which topological defects control many-body energy scales through their wavefunction structure, suggesting a general principle for engineering many-body energy scales via topology.

Figures (6)

• Figure 1: Schematic of the solitonic Kondo system. The Dirac system with a mass that changes the sign at $x=0$ from $-m$ to $m$ is considered (Fig. (a)). At the position where the mass changes sign, a solitonic mode is localized with the localization length $\xi \sim 1/m$. In the localized mode, the strength of the Coulomb interaction $U$ becomes $U\propto gm$, which inhibits the double occupation in the solitonic mode and results in the formation of a local moment (Fig. (b)). If this system is attached to another bulk system, such as the substrate, the hybridization takes place and its strength $\Gamma$ highly depends on the shape of the soliton (Fig. (b)). This hybridization works as a low-pass filter and is thus called a topological filter. Due to the topological filter, the high energy region down from bandwidth $D$ to $m$ is cutoff and the relevant UV scale for the Kondo effect becomes $m$, resulting in a significant $m$ dependence of the Kondo temperature $T_{\rm K}$ (Fig. (d)).
• Figure 2: Solitonic mode as a function of $x$ (Fig.(a)) and topological filter emerged from the solitonic mode (Fig.(b)). The solitonic mode is localized in the vicinity of the domain wall placed at $x=0$. The topological mass controls the localization length as depicted in Fig.(a) (solid, dashed, and dotted lines correspond to $m=1$, $m=0.5$, and $m=2$, respectively). In Fig. (b), the topological filter $F(\epsilon, m)$ defined in Eq. \ref{['topfilter']} is plotted as a function of $\epsilon/m$. The rapid $\epsilon^{-4}$ decay for $\epsilon \gg m$ demonstrates that the high-energy itinerant states are effectively decoupled from the soliton spin, allowing the topological mass $m$ to function as a physical UV cutoff for the Kondo scaling.
• Figure 3: Topological control of the Kondo temperature. Kondo temperature $T_{\rm K}$ is plotted as a function of the topological mass $m$ for $g=2.0$ and $\rho V^2 = 0.15$ (Fig. (a)). The numerical results obtained by scaling flow that starts from the initial coupling $g_0=\rho J(m)$ (black solid line) are in excellent agreement with the analytical formula $T_K \approx \mathcal{C} m \exp(-Am^2)$ (red dashed line) derived in Eq. \ref{['TKapproximated']}. The prefactor $\mathcal{C}$ is expected to depend on subleading corrections beyond the one-loop scaling analysis and is therefore treated as a non-universal constant. The characteristic dome-shaped behavior emerges from the competition between the localization-induced enhancement of the effective interaction $U_{\text{eff}} \propto m$ and the suppression of the hybridization volume. The blue dotted line $T_{\rm K} = m$ indicates the theoretical upper bound; beyond this limit (where $T_{\rm K} \approx m$), the system is expected to crossover into a valence-fluctuation regime. Figure (b) shows $\log(T_{\rm K}/m)$ as a function of $m^2$ (solid line). The dashed line is the linear fit of the data (here the slope is $\sim -2.193$).
• Figure 4: A double isomeric Class-II oligo (indenoindene) as an SSH model with a strong Coulomb interaction. Due to the difference in the distance, $t_2> t_1$ is naturally realized.
• Figure 5: Topological filter as a function of $\epsilon/\overline{m}$ (Fig. (a)) and Kondo temperature $T_{\rm K}$ as a function of the scaled topological mass (Fig. (b)). The black dots represent the numerical scaling flow that starts from the initial coupling $g_0=\rho J(m)$ for $g=2.0$ and $\rho V^2 = 0.15$, where we assume the dispersion relation in the bulk is given as $\epsilon=B^2k^2$ and define the scaled topological mass as ${\overline m}=Bm$. The red solid line shows the approximated expression given in Eq. \ref{['TKapproximated']}, and the dimensionless constant is chosen such that the peak value coincides with the numerical result.