  We search for lepton-flavor-violating $\tau^-\to e^-\alpha$ and
$\tau^-\to\mu^-\alpha$ decays, where $\alpha$ is an invisible spin-0 boson. The
search uses electron-positron collisions at $10.58$ GeV center-of-mass energy
with an integrated luminosity of $62.8$ fb$^{-1}$, produced by the SuperKEKB
collider and collected with the Belle II detector. We search for an excess in
the lepton-energy spectrum of the known $\tau^-\to e^-\bar\nu_e \nu_\tau$ and
$\tau^-\to \mu^-\bar\nu_\mu \nu_\tau$ decays. We report 95\% confidence-level
upper limits on the branching-fraction ratio ${\mathcal B}(\tau^-\to e^-
\alpha) / {\mathcal B}(\tau^- \to e^- \bar\nu_e \nu_\tau)$ in the range
$(1.1-9.7) \times 10^{-3}$ and on ${\mathcal B}(\tau^-\to \mu^- \alpha) /
{\mathcal B}(\tau^- \to \mu^- \bar\nu_\mu \nu_\tau)$ in the range $(0.7-12.2)
\times 10^{-3}$ for $\alpha$ masses between 0 and 1.6 GeV/c$^2$. These results
provide the most stringent bounds on invisible boson production from $\tau$
decays.
