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Bogomol'nyi Equations in Two-Species Born--Infeld Theories Governing Vortices and Antivortices

We derive several new Bogomol'nyi (self-dual) equations in two-species $U(1)\times U(1)$ gauge theories governed by the Born--Infeld nonlinear electrodynamics. By identifying appropriate Born--Infeld type Higgs potentials, we show that the highly nonlinear energy functionals admit exact topological lower bounds saturated by coupled first-order equations. The resulting models accommodate both vortex-vortex and vortex-antivortex configurations and generalize previously known single-species Born--Infeld systems to interacting multi-component settings. Beyond the derivation of the Bogomol'nyi equations, we develop an exact thermodynamic theory for pinned multivortex configurations in both the full plane and compact doubly periodic domains. Owing to the linear dependence of the Bogomol'nyi energy spectrum on topological charges, we obtain closed-form expressions for the canonical partition function, internal energy, heat capacity, and magnetization. In compact domains, the Bradlow type geometric bounds constrain admissible vortex numbers and lead to qualitatively new high-temperature behavior. In particular, vortex-only systems exhibit spontaneous magnetization, while vortex-antivortex systems do not, reflecting the underlying symmetry between opposite topological charges. These results provide a rare analytically solvable framework for studying thermodynamics in nonlinear multi-component gauge theories regulated by the Born--Infeld electrodynamics.