  Context. Extended gamma-ray TeV emission (TeV halos) around middle-aged
pulsars has been detected. A proposed model to explain these TeV halos is that
electrons from a degree-wide Pulsar Wind Nebula (PWN) get up-scattered by
cosmic microwave background photons through inverse Compton processes. However,
no X-ray degree-wide faint diffuse PWNe have been found around these
middle-aged pulsars in previous X-ray observations.
  Aims. We have performed a search for degree wide PWNe around Geminga, PSR
B0656+14, B0540+23, J0633+0632, and J0631+1036, using data from the first four
consecutive Spectrum Roentgen Gamma/eROSITA all-sky surveys. In order to better
understand the mechanisms underlying the formation of TeV halos, we
investigated the magnetic field strength in the degree wide neighbourhood of
those pulsars.
  Results. We did not detect degree-wide diffuse emission around Geminga, PSR
B0656+14, B0540+23, J0633+0632, and J0631+1036, which can be attributed to
being powered by the rotation-powered pulsars. Indeed, a close inspection of
the data shows that the pulsars of interest are all embedded in diffuse
emission from supernova remnants like the Monogem Ring or the Rosetta Nebula,
while PSR B0540+23 is located ~2.5 degrees away from the bright Crab pulsar,
which shines out the eROSITA point-spread function up to the position of PSR
B0540+23 and thus reduced the sensitivity to search for degree wide bright
diffuse X-ray emission strongly.
  Conclusions. Despite the non-detection of any degree-wide PWN surrounding the
analysed pulsars, we set flux upper limits to provide useful information on
magnetic field strength and its spatial distribution around those pulsars,
providing additional constraints to the proposed theory for the formation of
TeV halos around pulsars.
