Non-commutative Dynamic Approaches to the Kibble-Zurek Scaling Limit with an Initial Gapless Order

Nonequilibrium many-body physics is one of the core problems in modern physics, while the dynamical scaling from a gapless phase to the critical point is a most important challenge with very few knowledge so far. In the driven dynamics with a tuning rate $R$ across the quantum critical point (QCP) of a system with size $L$, the finite-time scaling shows that the square of the order parameter $m^2$ obeys a simple scaling relation $m^2\propto R^{2\beta \mathrm{/}\nu r}$ in the Kibble-Zurek (KZ) scaling limit with $R{L}^r\gg 1$. Here, by studying the driven critical dynamics from a gapless ordered phase in the bilayer Heisenberg model, we unveil that the approaches to the scaling region dominated by the KZ scaling limit with $R{L}^r\gg 1$ are {\it non-commutative}: this scaling region is inaccessible for large $R$ and finite medium $L$, while merely accessible for large $L$ and moderately finite $R$. We attribute this to the memory effect induced by the finite-size correction in the gapless ordered phase. This non-commutative property makes $m^2$ still strongly depends on the system size and deviates from $m^2\propto R^{2\beta \mathrm{/}\nu r}$ even for large $R$. We further show that a similar correction applies to the imaginary-time relaxation dynamics. Our results establish an essential extension of nonequilibrium scaling theory with a gapless ordered initial state.