  The pulsar timings are sensitive to both the nanohertz gravitational-wave
background and the oscillation of ultralight dark matter. The Hellings-Downs
angular correlation curve provides a criterion to search for stochastic
gravitational-wave backgrounds at nanohertz via pulsar timing arrays. We study
the angular correlation of the timing residuals induced by the spin-2
ultralight dark matter, which is different from the usual Hellings-Downs
correlation. At a typical frequency, we show that the spin-2 ultralight dark
matter can give rise to the deformation of the Hellings-Downs correlation curve
induced by the stochastic gravitational wave background.
