Thermodynamics and Spin Polarizations of the Fractional Quantum Hall States

The filling factor and temperature dependence of the electron spin polarization $〈S_z\left(T\right)〉$ of a two-dimensional electron system have been studied for various values of Zeeman coupling. At $T=0\mathrm{}$, there are sharp spin transitions for all the filling fractions considered here except $\nu =\frac{1}{3}$. At low $T$, the appearance of a peak in $〈S_z\left(T\right)〉$ at $\nu =\frac{2}{3}$ and $\nu =\frac{2}{5}$ is interpreted as a manifestation of the reentrant behavior observed earlier. At $\nu =\frac{3}{5}$, one sees a gradual trasition from one spin state to another at low $T$. At large $T$, $〈S_z\left(T\right)〉$ generally decays as $T^{-1\mathrm{}}$. These can be explored in the optically pumped nuclear magnetic resonance Knight shift measurements.