Ruderman-Kittel-Kasuya-Yosida interactions on a bipartite lattice

Carrier-mediated exchange coupling, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, plays a fundamental role in itinerant ferromagnetism and has great application potentials in spintronics. A recent theorem based on the imaginary-time method shows that the oscillatory RKKY interaction becomes commensurate on bipartite lattice and predicts that the effective exchange coupling is always ferromagnetic for the same sublattice but antiferromagnetic for opposite sublattices. We revisit this important problem by real- and imaginary-time methods and find that the theorem misses important contributions from zero modes. To illustrate the importance of zero modes, we study the spin susceptibility in graphene nanoribbons numerically. The effective exchange coupling is largest on the edges but does not follow the predictions from the theorem.

Figure 1

(Color online) A zigzag graphene nanoribbon with width $L_y=12$, length $L_x$, and sublattices represented by squares and circles. Each site is defined by integral values of $x$ and $y$, e.g., the rectangle contains all sites with $x=2$.

Figure 2

(Color online) The static spin susceptibility $\chi \left(x,1;x^{\prime },y^{\prime }\right)$ for a zigzag graphene nanoribbon at zero temperature. The edge response dominates and so we choose $y^{\prime }=1,L_y$ for a symmetric nanoribbon $L_y=12$—(a) at half-filling $\mu =0$, (b) at slight doping $\mu =0.01$—and a asymmetric nanoribbon $L_y=10$—(c) at half-filling $\mu =0$, (d) at slight doping $\mu =0.01$.