Speaker
Description
Operator scrambling is usually diagnosed through the growth of out-of-time-ordered correlators (OTOCs), yet a general symmetry principle underlying their effective dynamics has remained elusive. We develop a symmetry-based effective field theory for operator scrambling, organized by a strong-to-weak U(1) symmetry breaking in operator space. The key observation is that the four-fold Keldysh contour representation of the OTOC admits, in the noninteracting fermion limit, an emergent strong U(1) symmetry in a doubled Hilbert-space description, even though the original system need not possess any ordinary conserved quantity. The associated slow mode is the phase of the strong-charge creation operator, whose conjugate density is identified with the local operator size. Generic interactions explicitly break the strong symmetry and generate a mass term for the would-be Goldstone mode, thereby turning diffusive operator spreading into chaotic growth. We further show that this mass term is strongly constrained by an emergent duality combining time reversal with contour permutation. This duality fixes the effective action up to quadratic order in the response field, ties the multiplicative noise strength directly to the Lyapunov exponent, and makes the positivity of the latter a consequence of convergence of the real-time path integral. The OTOC dynamics is then governed by a noisy Fisher--Kolmogorov--Petrovsky--Piskunov equation, capturing in a unified framework the early exponential growth, ballistic propagation, nonlinear saturation, and stochastic front broadening of operator scrambling. We verify the construction in a Brownian Sachdev-Ye-Kitaev chain, where a direct saddle-point expansion reproduces the symmetry-based effective action. Our results reveal a symmetry origin of operator-size hydrodynamics and provide a principled route to effective theories of quantum scrambling beyond model-specific master equations.