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

Abstract

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 $m^2\propto R^{2β/νr}$ in the Kibble-Zurek (KZ) scaling limit with $RL^r\gg1$. 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 $RL^r\gg1$ 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 $m^2\propto R^{2β/νr}$ 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.

Figures (10)

• Figure 1: Bilayer Heisenberg model. (a) There are two different couplings: $J$ (intraplane) and $J_{\perp}$ (interplane), as indicated. (b) Fixing $J_{\perp} = 2.522$, the system is in the gapped dimer phase when $J < 1.0$, and in the gapless antiferromagnetic (AFM) phase when $J> 1.0$. The blue arrow represents the driving starting from the AFM state.
• Figure 2: Bilayer Heisenberg model: driven dynamics of $m^2$ versus $R$ for different system size $L$ on log-log plots. From the gapless AFM phase to the QCP using linear protocol before (a) and after rescaling (b). The black solid line segments in (a) have slopes of $2\beta/(\nu r)$, while the black dashed lines have slopes of $2\beta/\nu r_{02}$. The red and blue curves are fitting lines corresponding to the three largest system sizes, obtained using our newly proposed scaling relation. The inset in (a) shows $m^2 \propto 0.22\,L^{-1} + 0.012$ for $R = 0.0045$ (black curve).
• Figure 3: Relaxation dynamics of $m^2$ versus $\tau$ for different system size $L$ on log-log plots. For a sudden quench from the gapless AFM phase to the QCP before (a) and after rescaling (b), the black solid line in (a) follows the conventional scaling $m^2 \propto \tau^{2\beta/(z\nu)}$, while the black dashed lines is given by the modified scaling $m^2 \propto \tau^{2\beta/(z_{\text{02}}\nu)}$. The red, blue and black curves are fitting lines obtained using our newly proposed scaling formula.
• Figure S1: Bilayer Heisenberg model: driven dynamics of $m^2$ versus $R$ for different system size $L$ on log-log plots. From the dimer phase to the QCP using linear protocol before (a) and after rescaling (b). The red line segments in (a) have slopes of ($2\beta-d\nu$)/($\nu r$).
• Figure S2: Relaxation dynamics of $m^2$ versus $\tau$ for different system size $L$ on log-log plots. For a sudden quench from the gapped dimer initial state to the QCP before (a) and after rescaling (b), the red line in (a) is described by the scaling relation $m^2 \propto \tau^{(d - 2\beta/\nu)/z}$.