Effective field theory for superconducting phase transitions

A bstract Employing the Schwinger-Keldysh formalism, we formulate an effective field theory for s-wave superconducting phase transition, where the dynamical variables consist of electromagnetic gauge field and complex scalar order parameter. Symmetry-constrained effective action allows systematic handling of dissipations and fluctuations. In particular, we explore the physical implications of higher-order terms, including those involving additional dynamical fields as well as higher time derivatives, on the real-time dynamics near the superconducting critical point. When appropriately truncated, the effective field theory reproduces the phenomenological Ginzburg-Landau equations. Upon crossing the critical temperature into the low-temperature phase, the electromagnetic gauge symmetry undergoes spontaneous breaking induced by the condensate of the order parameter. Collective excitation analysis reveals that the Higgs mode behaves as an overdamped diffusive mode near the critical point, while the phase fluctuation is absorbed into the gauge field via the Higgs mechanism. For complex kinetic coefficients, we derive the coupled linearized equations for amplitude, phase, and longitudinal gauge fluctuations, and distinguish propagation from damping through their complex poles. The holographic Schwinger-Keldysh construction provides an explicit realization and a nontrivial consistency check of the effective action, and determines its Wilsonian coefficients within a specific strongly coupled model. Holographic results reveal a complex relaxation parameter that indicates oscillatory dynamics characteristic of strongly coupled systems.