  We construct effective field theory for SU(2) isospin charge diffusion, based
on holographic Schwinger-Keldysh contour arXiv:2008.01269. The holographic
model consists of a probe SU(2) gauge field in a doubled Schwarzschild-AdS$_5$
geometry. Accurate to first order in derivative expansion, we analytically
compute the effective action up to quartic order in hydrodynamical fields. The
effective theory contains both non-Gaussianity for noises and nonlinear
interactions between noises and dynamical variables. Moreover, the effective
theory captures both thermal and quantum fluctuations, which perfectly satisfy
dynamical Kubo-Martin-Schwinger (KMS) symmetry at quantum level. Interestingly,
the dynamical KMS symmetry, which is crucial in formulating non-equilibrium
effective field theory for a quantum many-body system, is found to have a nice
holographic interpretation.
