Hysteresis effects in the phase diagram of multiferroic GdMnO${}_3$

We present high-resolution thermal expansion $\alpha (T)$ and magnetostriction $\mathrm{\Delta }L(H)\mathrm{/}L$ measurements of GdMnO${}_3$, which develops an incommensurate antiferromagnetic order (ICAFM) below $T_{\mathrm{N}}\simeq$ 42 K and transforms into a canted A-type antiferromagnet (cAFM) below $T_{\mathrm{c}}\simeq 20$K. In addition, a ferroelectric polarization $\mathbf{P}\mathrm{\mid }\mathrm{\mid }a$ is observed below $T_{\rm FE} $forfinitemagneticfieldsappliedalongthe$b$ direction. In zero magnetic field we find a strongly anisotropic thermal expansion with certain, rather broad anomalous features. In finite magnetic fields, however, very strong anomalies arise at $T_{\mathrm{c}}$ for fields applied along each of the orthorhombic axes and at $T_{\mathrm{F}\mathrm{E}}$ for fields along the b axis. Both phase transitions are of first-order type and strongly hysteretic. We observe a down-bending of the ICAFM-to-cAFM phase boundary $T_{\mathrm{c}}(H)$ for low magnetic fields and our data give evidence for coexisting phases in the low-field low-temperature range.