Heisenberg necklace model in a magnetic field

We study the low-energy sector of the Heisenberg necklace model. Using the field-theory methods, we estimate how the coupling of the electronic spins with the paramagnetic Kondo spins affects the overall spin dynamics and evaluate its dependence on a magnetic field. We are motivated by the experimental realizations of the spin-1/2 Heisenberg chains in ${\mathrm{SrCuO}}_2$ and ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$ cuprates, which remain one-dimensional Luttinger liquids down to temperatures much lower than the in-chain exchange coupling $J$. We consider the perturbation of the energy spectrum caused by the interaction $\gamma$ with nuclear spins ($I=3/2$) present on the same sites. We find that the resulting necklace model has a characteristic energy scale, $\mathrm{\Lambda }\sim J^{1/3}{\left(\gamma I\right)}^{2/3}$, at which the coupling between (nuclear) spins of the necklace and the spins of the Heisenberg chain becomes strong. This energy scale is insensitive to a magnetic field $B$. For ${\mu }_{B}B>\mathrm{\Lambda }$ we find two gapless bosonic modes that have different velocities, whose ratio at strong fields approaches a universal number, $\sqrt{2}+1$.