Universal Short-Imaginary-Time Quantum Critical Dynamics Near Boundaries

While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter $M_s$ decays with imaginary time $\tau$ as $M_s \propto \tau^{-\beta_1/\nu z}$, where $\beta_1$ denotes the boundary order parameter exponent, and $\nu$ and $z$ correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent $\theta_1$, which is closely related to the critical initial slip behavior of $M_s$ characterized by the corresponding exponent $\theta_1'$. In contrast to its positive bulk counterpart, the boundary initial-slip exponent $\theta_1'$ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of $d$-dimensional quantum phase transitions generally coincide with those of $(d+1)$-dimensional classical phase transitions, we show that $\theta_1$ does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.