Abstract
Under holographic prescription for Schwinger-Keldysh closed time contour for non-equilibrium system, we consider fluctuation effect of the order parameter in a holographic superconductor model. Near the critical point, we derive the time-dependent Ginzburg-Landau effective action governing dynamics of the fluctuating order parameter. In a semi-analytical approach, the time-dependent Ginzburg-Landau action is computed up to quartic order of the fluctuating order parameter, and first order in time derivative.
Highlights
JHEP09(2021)168 latter is called Schwinger-Keldysh (SK) formalism [3, 4], which makes the descriptions of quantum systems in and out of equilibrium unified, and becomes an ideal framework for studying real-time dynamics
The effective action can be split into three parts: the time-dependent Ginzburg-Landau effective action SGL, which is the real-time generalization of HGL (1.1); the normal current part SN, which describes the dynamics of the charge diffusion; and the supercurrent part SS, which is responsible for the interaction between the order parameter and the external gauge field
We consider a holographic superconductor model [22, 23], in which spontaneously breaking of boundary U(1) symmetry is realized as formation of scalar hair outside the event horizon of Schwarzschild-AdS black hole [24]
Summary
I.e. without considering the backreaction of matter fields, a holographic model for s-wave superconductor is the scalar QED in Schwarzschild-AdS geometry [22, 24]. In appendix A, starting from bulk path integral (2.16), we demonstrate as long as dynamical EOMs are correctly taken (in compatible with a specific gauge choice), the partially on-shell bulk action S0|p.o.s computed in different gauge choices takes the same form in terms of slow modes Such an off-shell procedure allows possible violation of current conservation by fluctuations, which is an essential ingredient of effective action. With a specific gauge choice, the dynamical EOMs can fully determine profiles of the bulk fields, given sufficient boundary conditions This approach was first used to resum all-order derivatives in fluidgravity correspondence [60,61,62], and employed to derive hydrodynamic effective action. While (2.19) and (2.20) are equivalent once partially on-shell bulk solutions are plugged in, we find (2.19) is more convenient for practical calculations
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