  Pulsar halos (also termed 'TeV halo') are a new class of $\gamma$-ray sources
in Galaxy, which manifest as extended $\gamma$-ray emission around middle-age
pulsars, as discovered around the Geminga pulsar, the Monogem pulsar and
PSR~J0622+3749 by HAWC and LHAASO. A consensus has been reached that the TeV
emission comes from the inverse Compton scattering of escaping
electrons/positrons from the PWN off soft background radiation field, while the
particle transport mechanism in the halo is still in dispute. Currently, there
are mainly three interpretations, namely, the isotropic, suppressed diffusion
model; the isotropic, unsuppressed diffusion model with considering ballistic
propagation of newly injected particles; the anisotropic diffusion model. While
the predicted gamma-ray surface brightness profiles by all three models can be
more or less consistent with the observation, the implication of the three
models for cosmic-ray transport mechanisms and the properties of interstellar
magnetic field are quite different. In this study, we calculate the anticipated
X-ray emission of pulsar halos under the three models. We show that the
synchrotron radiation of these escaping electrons can produce a corresponding
X-ray halo around the pulsar, and the expected surface brightness profiles are
distinct in three models. We suggest that sensitive X-ray detectors of a large
field of view (such as eROSITA and Einstein Probe) with a reasonably long
exposure time are crucial to understand the formation mechanism of pulsar halos
and serve as a probe to the properties of the interstellar turbulence.
