Peculiarities of theO (n)model forn<1

Abstract

The $O \mathrm{}\left(n\right)\mathrm{}$ model consisting of $n$-component spins $\overrightarrow{S}$ with the constraint ${\overrightarrow{S}}^2=\lambda$ in a magnetic field $h$ is studied. It is shown that mathematically it is possible for the susceptibility to become negative for $n<\mathrm{}1\mathrm{}$, which implies a violation of convexity properties for $n<\mathrm{}1\mathrm{}$. In a mean-field approximation, the susceptibility ${\chi }_n$ and the specific heat $C_n$ are positive near the critical temperature $T_c$ for all $n\ge 0$ in contradiction with the $\epsilon$ expansion, but they become negative at very low temperatures for $n<\mathrm{}1\mathrm{}$. It is also shown that the spontaneous magnetization is not a monotone function of temperature for $n<\mathrm{}1\mathrm{}$. Our calculation also supports the conclusion drawn by des Cloizeaux that the low-temperature phase of the $O \mathrm{}\left(0\right)\mathrm{}$ model describes the semidilute regime of the polymer system as $h\to 0$.