RKKY interaction in Weyl semimetal nanowires

TL;DR

This work analyzes how magnetic impurities on the surface of a Weyl semimetal nanowire interact via RKKY exchange, mediated by both Fermi-arc surface states and low-energy bulk modes. Using a Green's-function formalism in a cylindrical geometry, the authors derive an effective spin Hamiltonian with anisotropic Heisenberg exchange, Dzyaloshinskii–Moriya terms, and a symmetric off-diagonal component, and they disentangle the distinct surface and bulk contributions. They find that FA surface states induce a $1/(z)^{2}$ decay with selective nonzero couplings (notably $J_{zz}$ and $J_{2}$), while bulk states slow the decay to $1/(z)^{ta}$ with $1<ta<2$, preserving overall symmetry determined by cylindrical geometry. These results position WSM nanowires as a versatile platform for engineering spin models and exploring spin textures, with implications for spintronics and topological-device applications.

Abstract

We investigate the effective couplings induced between localized impurities on the surface of a Weyl semimetal (WSM) nanowire within the framework of Ruderman--Kittel--Kasuya--Yosida (RKKY) theory. The itinerant electrons from the chiral Fermi arc surface states mediate impurity-impurity interaction at low energies. As a result, the spin-momentum locking naturally plays a central role in shaping the spin-spin correlations. We show that the dominant interaction channels have distinct origins: while the azimuthal coupling, $J_{φφ}$ term arises exclusively from Fermi arc states with identical spin polarization, the couplings $J_{μν}$ ($μ,ν= z,r$) are governed by Fermi arc states with opposite spin polarizations. Furthermore, we demonstrate that purely surface-mediated contributions exhibit different scaling behavior compared to those involving Fermi arcs and low-energy bulk states. We systematically untangle the contributions from bulk and surface states to the RKKY couplings, using analytical and numerical methods. Our results establish WSM nanowires as a versatile platform for engineering and simulating a broad class of spin models.

Figures (5)

• Figure 1: Schematic setup for the Weyl-semimetal cylindrical nanowire of radius $R$ and infinite length along the $\hat{z}$ direction. Impurity spins $\bold{S_{1}}$ and $\bold{S_{2}}$ are placed on the surface of the cylinder, and they can interact via the conduction electrons in the Weyl semimetal.
• Figure 2: Top: WSM dispersion in nanowire geometry as a function of $k_{z}$. There are four Weyl nodes (located at $k_{z}=\pm k_{0}\pm \pi/2$, where $2k_{0}$ represents the length of each Fermi arc and $\pi$ is the separation between the center of the two Fermi arcs in momentum space) in bulk and two Fermi-arcs joining them. The two blue dots corresponds to the location of Weyl nodes at $\pm k_{z}$.The gap in the lowest energy band is proportional to the inverse of the radius $R$ of the nanowire. Bottom: Spin densities of the Fermi-arc surface states are shown as a function of radial distance $r$ for two values of $k_{z}$. The FA surface states are polarized along the $\hat{\phi}$ direction with $\langle \hat{\sigma}_{z}\rangle =\langle \hat{\sigma}_{r}\rangle=0$ and Fermi arcs with $k_{z}=\pm 1.44$ have exactly opposite spin polarization. The localization of the surface states depends primarily on the radius of the cylinder. All the lengths are measured in the unit of the lattice constant $a$, for our convenience we set $a=1$. Other parameters for the plot are as follows: $R$=25.0, $k_{0}=0.2\pi$, $v_{F}=1$.
• Figure 3: We plot all the distinct effective spin-spin correlation between impurity spins mediated by conduction electrons on Weyl-semimetal nano-wire as a function of $\delta z$ for fixed $\delta \phi$ in cylindrical coordinate ($\delta z=z_{1}-z_{2}, \delta \phi=\phi_{1}-\phi_{2}$). The symmetries found from the numerical simulation are as follows: $J_{zz}=J_{rr}$, $J_{rz}=-J_{zr}$, $J_{r \phi}=-J_{\phi r}$, $J_{z \phi}=J_{\phi z}$. The magnitude of the correlations $J_{r \phi}, J_{\phi r}, J_{z \phi}, J_{\phi z}$ are very small compared to the other correlations that are shown here. Here, the RKKY couplings depend on three frequencies which are determined primarily by two scales, namely the Fermi energy and the width of each Fermi arc. Also, in the parameter range used here, the coupling $J_{\phi \phi}$ is predominately antiferromagnetic, ferromagnetic for small and large $\delta z$ respectively. On the top panel the Fermi energy is set at $\epsilon_{F}=0.44$, and on the bottom panel $\epsilon_{F}=0.64$. The Fermi energy is set to values that include contributions from both Fermi arcs and low-energy bulk states. Other parameters are as follows : $R=7$, $\delta \phi=0.1$, $k_{0}=0.05 \pi$, $|m_{\text{max}}|=9$. All the correlations are measured in the unit of $10^{-3} J^{2}$. Numerical convergence are upto order $10^{-7} J^{2}$.
• Figure 4: In this figure we have compared the distinct RKKY correlations as a function of $\delta \phi$ for fixed $\delta z=2.0$, $\epsilon_{F}=0.44$. Other parameters are the same as Fig. (\ref{['wsmcorr1']}). RKKY correlation $J_{zz}$ and $J_{\phi \phi}$ symmetric around $\delta \phi =\pi$ but, $J_{zr}$ shows antisymmetric dependence. All the correlations are measured in the unit of $10^{-3} J^{2}$. Numerical convergence are atleast $10^{-7} J^{2}$.
• Figure A1: The correlation $J_{\phi z},J_{z \phi}, J_{r \phi}, J_{\phi r}$ are plotted as a function of $\delta z$, with $\epsilon_{F}=0.44$, all the other parameter is same as in Fig (\ref{['wsmcorr1']}). These correlations become zero analytically when we take only the contributions coming from the Fermi-arc surface states.