Achieving the Landau bound to precision of quantum thermometry in systems with vanishing gap

We address estimation of temperature for finite quantum systems at thermal equilibrium and show that the Landau bound to precision δT2 ∝ T2, originally derived for a classical not too small system being a portion of a large isolated system at thermal equilibrium, may be also achieved by energy measurement in microscopic quantum systems exhibiting vanishing gap as a function of some control parameter. On the contrary, for any quantum system with a nonvanishing gap Δ, precision of any temperature estimator diverges as δT2 ≳ T4eΔ/T.