Heisenberg Necklace Model in Magnetic Field

Motivated by the experimental realizations of nearly one-dimensional spin-1/2 Heisenberg model found in chain cuprates SrCuO$_2$ and Sr$_2$CuO$_3$, which remain in a quantum-critical Luttinger liquid state down to temperatures that are much lower than in-chain exchange coupling, we consider the perturbation to this state caused by interactions with nuclear spins on the same sites. We study the low-energy sector of the Heisenberg Necklace model and estimate the effect of the coupling between the nuclear and the electronic spins on the overall spins dynamics and its dependence on the magnetic field. We find that the Necklace model has a characteristic energy scale, $\Lambda \sim J^{1/3}(\gamma I)^{2/3}$, at which the coupling between spins of the necklace and the spins of the Heisenberg chain becomes strong. In the strong magnetic field $\mu_B B > \Lambda$ the low energy spectrum is gapless, but two gapless bosonic modes have different velocities whose ratio at strong fields approaches a universal number, $\sqrt 2 +1$. In the case of Sr$_2$CuO$_3$ the energy scale $\Lambda $ is sizable and comparable to the Neel ordering temperature induced by the inter-chain coupling, and thus could noticeably modify the low temperature magnon dynamics. We further find that the above energy scale is insensitive to strong magnetic field, $\mu_B B \gg \Lambda \sim J^{1/3}(\gamma I)^{2/3}$, and therefore the interaction with nuclear spins cannot lead to unusually strong magnetic field dependence of the magnon spectrum observed by ESR in Sr$_2$CuO$_3$, which has been attributed to the magnon interaction with the Higgs mode.