Dynamical self-dual criticality in Fibonacci-monitored quantum Ising chains

For the quantum phase transition in the transverse-field Ising chain, Kramers-Wannier duality not only protects its critical properties but also pinpoints the location of the phase transition. Its role in out-of-equilibrium, monitored dynamics, however, remains largely unexplored beyond time-periodic Floquet protocols where self-duality turns into a statistical average symmetry. Here we explore the emergence of dynamical self-duality in the absence of time-translation symmetry by investigating the monitored dynamics of one-dimensional Ising/Majorana chains where measurements are arranged in a quasiperiodic Fibonacci sequence. We find that the dynamical extension of this non-invertible symmetry to an out-of-equilibrium setting allows one to organize the dynamical phase diagram of entangled phases, both predicting the transition locations and protecting universal critical behavior. Analytically and numerically, we identify two distinct critical lines, both related to the golden ratio, for Born-rule weak measurements and for random Clifford projective measurements. The latter coincides with the transition of a pure imaginary-time evolution, which can be viewed as a post-selected trajectory. The universality classes of the long-time critical steady states at Fibonacci times are determined, while the transient dynamics between Fibonacci times is deformed by measurements, realizing dynamical measurement-altered quantum criticality in real time.