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

Two-dimensional confinement of critical U(1) gauge field in the presence of Fermi surface