A quantum spherical spin model with an antiferromagnetic coupling $J$ on the $A{B}_2$ chain is studied, whose topology is of interest in the context of ferrimagnetic polymers and oxocuprates. A ferrimagnetic long-range order is found at the only critical point $g=T=h=0$, where $T$ denotes the temperature, $h$ the magnetic field, and $g$ the quantum coupling constant in energy units. The approach to the critical point, with diverging correlation length $\xi$ and cell susceptibility ${\chi }_{\text{cell}}$, is characterized through several paths in the $\{g,T,h\}$ parameter space: for $(T∕J)\to 0$ and $g=h=0$, ${\chi }_{\text{cell}}\sim 1∕T^2$, as also found in several classical and quantum spherical and Heisenberg models; for $(h∕J)\to 0$ and $g=T=0$, ${\chi }_{\text{cell}}\sim 1∕h$; and for $(g∕J)\to 0$ and $T=h=0$, $\ln {\chi }_{\text{cell}}\sim 1∕\sqrt{g}$, thus evidencing an essential singularity due to quantum fluctuations. In any path chosen the relation ${\chi }_{\text{cell}}\sim {\xi }^2$ is satisfied. For finite $g$ and $T$ a field-induced short-range ferrimagnetism occurs to some extent in the $\{g,T,h\}$ space, as confirmed by the analysis of the local spin averages, cell magnetization with a rapid increase for very low fields, and spin-spin correlation functions. The asymptotic limits of the correlation functions are also provided with respect to $g$, $T$, $h$, and spin distance. The analysis of the entropy and specific heat reveals that the quantum fluctuations fix the well-known drawback of classical spherical models concerning the third law of thermodynamics.

• Received 5 August 2005

DOI:https://doi.org/10.1103/PhysRevB.72.214420