Equilibrium and real-time properties of the spin correlation function in the two-impurity Kondo model

Abstract

We investigate the equilibrium and real-time properties of the spin-correlation function $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$ in the two-impurity Kondo model for different distances $R$ between the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$. For distances $R$ exceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings $J$ between the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain length scale introduced by finite temperature. The real-time dynamics after a sudden quench of the system reveals that correlations propagate through the conduction band with Fermi velocity. We identify two distinct timescales for the long-time behavior, which reflects that for small $J$ the system is driven by the RKKY interaction while for large $J$ the Kondo effect dominates. Interestingly, we find that at certain distances a one-dimensional dispersion obeying $\ensuremath{\epsilon}(k)=\ensuremath{\epsilon}(\ensuremath{-}k)$ may lead to a local parity conservation of the impurities such that $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$ becomes a conserved quantity for long times and does not decay to its equilibrium value.