  The compact U(1) gauge field occurs in many fractionalized descriptions of
low dimensional quantum magnetism and heavy fermion systems. In this respect a
fundamental question about the gauge field is whether it is confined or not in
the presence of Fermi surface. In this paper we show that the U(1) gauge field
with a possible positive anomalous dimension in the presence of Fermi surface
has to be in the confined state, although it is strongly coupled to abundant
gapless fermionic excitations near the Fermi surface in two spatial dimensions.
This means that some quantum spin liquids and/or fractionalized metallic states
described in terms of this kind of gauge fields with positive anomalous
dimensions are unstable to other possible symmetry-breaking states, as a
result, they are not be appropriate candidates for ground states in related
systems. The instability obtained shows some novel features of the quantum spin
liquids and/or fractionalized metallic states in the presence of Fermi surface
which have not been reported in the literature. The result could be useful for
further study in the quantum spin liquids and many related slave-particle gauge
theories in strongly correlated electron systems.
