Dynamical discontinuities in repeated weak measurements revealed by complex weak values

Critical phenomena reveal universal behavior in complex systems, and uncovering analogous effects in quantum weak measurement protocols with post-selection provides new insight into how measurement backaction can shape quantum dynamics. This work investigates dynamical discontinuities that arise when the post-selected polar angle is used as a control parameter. The system evolves under repeated applications of a weak measurement protocol with post-selection, in which the meter state is retained after each iteration while the system is renewed. The emergence of these discontinuities is shown to be determined by the structure of the weak value: when the weak value has a nonzero imaginary component, a discontinuity appears in the expectation value of meter observables precisely at the point where the imaginary part of the weak value vanishes as a function of the post-selection polar angle. In contrast, no discontinuities occur when the weak value remains real for all post-selection angles. The phenomenon originates from the eigenstructure of the protocol's Kraus operator, with the stability of fixed points changing at the critical point where the discontinuity arises. Remarkably, the associated critical exponent is 1, independent of system parameters. These results open new perspectives for engineering non-analytic behavior in measurement-based quantum control and for probing criticality in post-selected quantum dynamics using weak measurements with weak values.