Chiral Spin Order in Kondo-Heisenberg Systems

We demonstrate that low dimensional Kondo-Heisenberg systems, consisting of itinerant electrons and localized magnetic moments (Kondo impurities), can be used as a principally new platform to realize scalar chiral spin order. The underlying physics is governed by a competition of the Ruderman-Kittel-Kosuya-Yosida (RKKY) indirect exchange interaction between the local moments with the direct Heisenberg one. When the direct exchange is weak and RKKY dominates, the isotropic system is in the disordered phase. A moderately large direct exchange leads to an Ising-type phase transition to the phase with chiral spin order. Our finding paves the way towards pioneering experimental realizations of the chiral spin liquid in systems with spontaneously broken time-reversal symmetry.

Figure 1

Competition between spin interactions in KHS. The spin on each lattice site is decomposed in terms of an orthonormal triad ${\mathit{e}}_{1,2,3}$ (green arrows) with ${\mathit{e}}_{\perp }=(-1{)}^{N(\mathit{r})}{\mathit{e}}_3$; see Eq. (3). The RKKY exchange interaction (red lines) is mediated by electrons (red circles) and favors helical-like configuration of the vectors ${\mathit{e}}_{1,2}$. The Heisenberg exchange interaction (blue line) favors antiparallel orientation of ${\mathit{e}}_{\perp }$ on neighboring lattice sites. Coupling constants $J_{K,H}$ are introduced in Eq. (2).

Figure 2

Chiral configuration of spins in the CSL phase. The dotted line is the helix. The green and red arrows show helical ${\mathit{S}}_{\parallel }$ and antiferromagnetic ${\mathit{S}}_{\perp }$ spin components, respectively; see Eq. (3). For simplicity, we disregard helix deformations on the scale of several lattice constants which are caused by the thermal fluctuations of the triad ${\mathit{e}}_{1,2,3}$.

Figure 3

Phase diagram of the isotropic 2D KHS on the plane $T$ vs $J_H$ at $T\ll s|J_K|$. The green line is the critical line; see Eq. (7). It separates the disordered phase and CSL (green area). $J_{\mathrm{AFM}}$ marks the transition from CSL to the antiferromagnetic phase (red line) at $T=0$. Inset: Temperature dependence of the mean value $⟨\sin (\alpha )⟩$. Note that the SCO parameter is proportional to this quantity; see Eq. (9).