Higher-order generalized hydrodynamics in one dimension: The noninteracting test

We derive a dynamical equation that describes the exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a (generalized) hydrodynamic theory. The question at hand is whether some large-time corrections are captured by higher-order hydrodynamics. We consider in particular the dynamics after two chains, prepared in different conditions, are joined together. In these situations, a light cone, separating regions with macroscopically different properties, emerges from the junction. In free fermionic systems some observables close to the light cone follow a universal behavior, known as Tracy-Widom scaling. Universality means a weak dependence on the system's details, so this is the perfect setting where hydrodynamics could emerge. For the transverse-field Ising chain and the $XX$ model, we show that hydrodynamics captures the scaling behavior close to the light cone. On the other hand, our numerical analysis suggests that hydrodynamics fails in more general models, whenever a condition is not satisfied.

Figure 1

The connected two-point function of ${\mathit{s}}^z$ as a function of the rescaled positions $x,y$ (18). The symbols are exact numerical data (in a chain with 1801 spins) at three different times, (a) $t=60,150,300$ and (b) $t=60,150,380$; after that, a thermal state with inverse temperature $\beta =1$ is put in contact with the infinite-temperature state. The lines are the predictions (20). (a) is for a generalized XY model with $J_{\ell ,0}^{yy}=-1$, $J_{\ell }^z=-2$, $J_{\ell ,1}^{xx}\approx -0.0796$, and $J_{\ell ,1}^{yy}\approx 0.3294$ [see (1)], for which $\alpha \approx 1$. (b) is for an $XY$ model with $J_{\ell ,0}^{xx}=0.15$, $J_{\ell ,0}^{yy}=-1.15$, and $J_{\ell }^z=-2$, for which $\alpha \approx 0.8$. (a) unveils an excellent agreement between data and predictions. On the other hand, in (b) the data do not seem to approach the predictions as the time is increased.