  Context. Coronal loops are the basic building blocks of the solar corona,
which are related to the mass supply and heating of solar plasmas in the
corona. However, their fundamental magnetic structures are still not well
understood. Most coronal loops do not expand significantly, whereas the
diverging magnetic field would have an expansion factor of about 5-10 over one
pressure scale height. Aims. In this study, we investigate a unique coronal
loop with a roughly constant cross section, it is ultra long and quite thin. A
coronal loop model with magnetic helicity is presented to explain the small
expansion of the loop width. Methods. This coronal loop was predominantly
detectable in the 171 A channel of the Atmospheric Imaging Assembly (AIA).
Then, the local magnetic field line was extrapolated by a
Potential-Field-Source-Surface model. Finally, the differential emission
measure analysis made from six AIA bandpasses was applied to obtain the thermal
properties of this loop. Results. This coronal loop has a projected length of
roughly 130 Mm, a width of about 1.5 +(-) 0.5 Mm and a lifetime of around 90
minutes. It follows an open magnetic field line. The cross section expanded
very little (i.e., 1.5-2.0) along the loop length during its whole lifetime.
This loop has a nearly constant temperature at about 0.7 +(-) 0.2 MK, whereas
its density exhibits the typical structure of a stratified atmosphere.
Conclusions. We use a thin twisted flux tube theory to construct a model for
this non-expanding loop, and find that indeed with sufficient twist a coronal
loop can attain equilibrium. However, we can not rule out other possibilities
such as footpoint heating by small-scale reconnection, elevated scale height by
a steady flow along the loop etc.
