  Within a (2+1)-dimensional U(1) gauge field theory, after calculating the
Dyson-Schwinger equation for fermion self-energy we find that chiral symmetry
breaking (CSB) occurs if the gauge boson has a very small mass but is
suppressed when the mass is larger than a critical value. In the CSB phase, the
fermion acquires a dynamically generated mass, which leads to antiferromagnetic
(AF) long-range order. Since in the superconducting (SC) state the gauge boson
acquires a finite mass via Anderson-Higgs mechanism, we obtain a field
theoretical description of the competition between the AF order and the SC
order. As a compromise of this competition, there is a coexistence of these two
orders in the bulk material of cuprate superconductors.
