  Temperature estimation of interacting quantum many-body systems is both a
challenging task and topic of interest in quantum metrology, given that
critical behavior at phase transitions can boost the metrological sensitivity.
Here we study non-invasive quantum thermometry of a finite, two-dimensional
Ising spin lattice based on measuring the non-Markovian dephasing dynamics of a
spin probe coupled to the lattice. We demonstrate a strong critical enhancement
of the achievable precision in terms of the quantum Fisher information, which
depends on the coupling range and the interrogation time. Our numerical
simulations are compared to instructive analytic results for the critical
scaling of the sensitivity in the Curie-Weiss model of a fully connected
lattice and to the mean-field description in the thermodynamic limit, both of
which fail to describe the critical spin fluctuations on the lattice the spin
probe is sensitive to. Phase metrology could thus help to investigate the
critical behaviour of finite many-body systems beyond the validity of
mean-field models.
