  Revealing the properties of single spin defects in solids is essential for
quantum applications based on solid-state systems. However, it is intractable
to investigate the temperature-dependent properties of single defects, due to
the low precision for single-defect measurements in contrast to defect
ensembles. Here we report that the temperature dependence of the Hamiltonian
parameters for single negatively charged nitrogen-vacancy (NV$^{-}$) centers in
diamond is precisely measured, and the results find a reasonable agreement with
first-principles calculations. Particularly, the hyperfine interactions with
randomly distributed $^{13}$C nuclear spins are clearly observed to vary with
temperature, and the relevant coefficients are measured with Hz-level
precision. The temperature-dependent behaviors are attributed to both thermal
expansion and lattice vibrations by first-principles calculations. Our results
pave the way for taking nuclear spins as more stable thermometers at nanoscale.
The methods developed here for high-precision measurements and first-principles
calculations can be further extended to other solid-state spin defects.
