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

TL;DR

This work derives six distinct Bogomol'nyi (self-dual) equations for two-species $U(1)\times U(1)$ gauge theories endowed with Born--Infeld electrodynamics, showing that each energy functional admits a topological lower bound saturated by coupled first-order equations for vortices and/or antivortices. The authors provide exact flux-quantization and minimum-energy expressions on both the full plane and compact doubly periodic domains, with compact-domain bounds (Bradlow bounds) depending explicitly on the Born parameters. Beyond the field-theoretic reductions, the paper develops an exact thermodynamic theory for pinned vortices, delivering closed-form partition functions and analytic expressions for internal energy, heat capacity, and magnetization, and revealing qualitatively distinct high-temperature behaviors between vortex-only and vortex–antivortex systems, as well as Meissner effects at zero temperature. The Born--Infeld nonlinearities, together with geometry, produce rich thermodynamic phases:Bradlow bounds regulate vortex accommodation on compact domains, yielding saturation or decay of magnetization in the high-temperature limit and distinguishing the vortex-only from vortex–antivortex ensembles. Overall, the results establish an analytically tractable framework linking nonlinear gauge dynamics, topology, and statistical mechanics in multi-component Born--Infeld theories, with potential relevance to multiband superconductors, cosmic strings, and related topological materials.

Abstract

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.