Quantum spherical spin model on the AB2 chain

Abstract

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 $\ensuremath{\xi}$ and cell susceptibility ${\ensuremath{\chi}}_{\text{cell}}$, is characterized through several paths in the ${g,T,h}$ parameter space: for $(T∕J)\ensuremath{\rightarrow}0$ and $g=h=0$, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕{T}^{2}$, as also found in several classical and quantum spherical and Heisenberg models; for $(h∕J)\ensuremath{\rightarrow}0$ and $g=T=0$, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕h$; and for $(g∕J)\ensuremath{\rightarrow}0$ and $T=h=0$, $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕\sqrt{g}$, thus evidencing an essential singularity due to quantum fluctuations. In any path chosen the relation ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}{\ensuremath{\xi}}^{2}$ 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.