  There is growing evidence that the superconducting semimetal FeSe ($T_c\sim8$
K) is in the crossover regime between weak-coupling Bardeen-Cooper-Schrieffer
(BCS) and strong-coupling Bose-Einstein-condensate (BEC) limits. We report on
longitudinal and transverse thermal conductivities, $\kappa_{xx}$ and
$\kappa_{xy}$, respectively, in magnetic fields up to 20 T. The field
dependences of $\kappa_{xx}$ and $\kappa_{xy}$ imply that a highly anisotropic
small superconducting gap forms at the electron Fermi-surface pocket whereas a
more isotropic and larger gap forms at the hole pocket. Below $\sim1.0$ K, both
$\kappa_{xx}$ and $\kappa_{xy}$ exhibit distinct anomalies (kinks) at the upper
critical field $H_{c2}$ and at a field $H^*$ slightly below $H_{c2}$. The
analysis of the thermal Hall angle ($\kappa_{xy}/\kappa_{xx}$) indicates a
change of the quasiparticle scattering rate at $H^*$. These results provide
strong support to the previous suggestion that above $H^*$ a distinct
field-induced superconducting phase emerges with an unprecedented large spin
imbalance.
