  In three-dimensional quantum electrodynamics (QED$_{3}$) with massive gauge
boson, we investigate the Dyson-Schwinger equation for the fermion self-energy
in the Landau gauge and find that chiral symmetry breaking (CSB) occurs when
the gauge boson mass $\xi$ is smaller than a finite critical value $\xi_{cv}$
but is suppressed when $\xi > \xi_{cv}$. We further show that the critical
value $\xi_{cv}$ does not qualitatively change after considering higher order
corrections from the wave function renormalization and vertex function. Based
on the relation between CSB and the gauge boson mass $\xi$, we give a field
theoretical description of the competing antiferromagnetic and superconducting
orders and, in particular, the coexistence of these two orders in high
temperature superconductors. When the gauge boson mass $\xi$ is generated via
instanton effect in a compact QED$_{3}$ of massless fermions, our result shows
that CSB coexists with instanton effect in a wide region of $\xi$, which can be
used to study the confinement-deconfinement phase transition.
