The ordering temperature $T_c$ of the easy-plane hexagonal antiferromagnet CsMn${\mathrm{F}}_3$ was measured as a function of magnetic field $H$, up to 120 kOe. $T_c$ was determined from the thermal expansion anomaly at constant $H$. At $H=0\mathrm{}$, $T_N\equiv T_c\mathrm{}\left(0\right)=51.4\mathrm{}$ K. When $\overrightarrow{\mathrm{H}}$ is in the hexagonal plane, the boundary $T_c\mathrm{}\left(H\right)\mathrm{}$ is bow shaped: with increasing $H$, $T_c$ first increases, then passes through a maximum, and later decreases. The maximum $T_c$ is ∼37 mK above $T_N$, and it occurs at $H\cong 29.5$ kOe. The bow-shaped phase boundary is attributed to the $\mathrm{XY}$-to-Ising crossover which is induced by the magnetic field, as discussed by Fisher, Nelson, and Kosterlitz. Fits to the phase boundary $T_c\mathrm{}\left(H\right)\mathrm{}$ give a crossover exponent $\phi =1.185\mathrm{}\pm 0.03$ for one sample and $\phi =1.184\mathrm{}\pm 0.025$ for another, compared to the theoretical value $\phi \mathrm{}(n=2\mathrm{})=1.175\mathrm{}\pm 0.015$. When $\overrightarrow{\mathrm{H}}$ is perpendicular to the hexagonal plane, $T_c$ decreases monotonically with increasing $H$, but the decrease is not in accordance with mean-field theory, which predicts a decrease proportional to $H^2$. The deviation from mean-field behavior is attributed to a virtual bicritical point (VBP) with Heisenberg symmetry, which exists mathematically at a negative value of $H^2$. Although the VBP cannot be observed directly, it affects the behavior in the observable region of $H^2>~0$. Physically, a magnetic field applied perpendicular to the easy plane enhances the Heisenberg-to-$\mathrm{XY}$ symmetry breaking, which at $H=0\mathrm{}$ is solely due to the weak easy-plane uniaxial anisotropy. The enhanced symmetry breaking causes a non-mean-field dependence of $T_c$ on $H$. An equation derived on this basis gives a good description of the phase boundary $T_c\mathrm{}\left(H\right)\mathrm{}$. This equation contains three adjustable parameters, two of which can also be estimated without recourse to the phase boundary $T_c\mathrm{}\left(H\right)\mathrm{}$. The values for these two parameters obtained from a best fit to $T_c\mathrm{}\left(H\right)\mathrm{}$ agree with the independent estimates.

• Received 17 September 1979

DOI:https://doi.org/10.1103/PhysRevB.21.1271