Critical behavior of the spontaneous magnetization at marginal dimensionality in LiHoF4

We have measured the critical behavior of the spontaneous magnetization as a function of temperature for $T<~T_c$ in the three-dimensional, uniaxial, dipolar-coupled ferromagnet LiHo${\mathrm{F}}_4$ using elastic-laser-light-scattering techniques. In the reduced-temperature range $1.3\times {10}^{-3\mathrm{}}<~t<~1.3\times {10}^{-1\mathrm{}}$, we find that the logarithmically corrected mean-field power law $\frac{M\left(T\right)\mathrm{}}{M\left(0\right)\mathrm{}}={\mathrm{Bt}}^{\frac{1}{2}}{\left|\ln \right|\frac{t_0}{t}\mathrm{}\left|\right|\mathrm{}}^{\frac{1}{3}}$ fits the data with $T_c=1.5383\mathrm{}\pm 0.0003$ K, $t_0=0.483\mathrm{}\pm 0.046$, and ${\chi }^2=1.34\mathrm{}$. We have also fitted the data to $\frac{M\left(T\right)\mathrm{}}{M\left(0\right)\mathrm{}}=B{\mathrm{}\left|t\right|\mathrm{}}^{\frac{1}{2}}{\left|\ln \mathrm{}\right|t\left|\right|}^{\frac{1}{3}}[1+C_1(\ln \frac{\left|\ln \mathrm{}\right|t\left|\right|}{\ln \mathrm{}\left|t\right|\mathrm{}})+C_2(\frac{1}{\ln \mathrm{}\left|t\right|\mathrm{}}\left)\right]$ in order to obtain explicitly the corrections to the leading singularities. We find in this fit that $T_c=1.5382\mathrm{}\pm 0.0003$ K, $C_1=0.07\mathrm{}\pm 0.3$, $C_2=0.287\mathrm{}\pm 0.07$, and ${\chi }^2=1.26\mathrm{}$. Fits to a simple power law $\frac{M\left(T\right)\mathrm{}}{M\left(0\right)\mathrm{}}={\mathrm{Bt}}^{\beta }$ yield $T_c=1.5370\mathrm{}\pm 0.0002$ K, $\beta =0.355\mathrm{}\pm 0.005$, and ${\chi }^2=17.2\mathrm{}$. We also present measurements of the temperature and magnetic-field dependence of the light scattering in LiHo${\mathrm{F}}_4$, and the magnetic-field and wavelength dependence of the circular birefringence.