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

Abstract

We derive a Moyal dynamical equation that describes exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a 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 that 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 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.Over the past few years, we are experiencing an increasing interest in the physics behind the nonequilibrium time evolution of inhomogeneous states. An example is the time evolution of two semi-infinite chains that are joined together after having been prepared in different equilibrium conditions [1,2]. This kind of settings allows one to investigate the transport properties of quantum many-body systems even if the system is isolated from the environment.The first analytic results in this context were obtained in noninteracting models . There, under the assumption of quasistationarity, a semiclassical picture applies where the information about the initial state is carried by free stable quasiparticles moving throughout the system. Similar results were obtained in the framework of conformal field theory and Luttinger liquid descriptions [25][26][27][28][29][30][31][32][33][34][35][36][37]. In the presence of interactions the situation was less clear [38][39][40][41][42][43][44][45][46][47], but, eventually, Refs [48,49] have shown that the continuity equations satisfied by the (quasi)local conserved quantities are sufficient to characterize the late-time behavior. The framework developed in Refs [48,49] is now known as generalized hydrodynamics [48], where "generalized" is used to emphasize that integrable models have infinitely many (quasi)local charges [50]. We will generally omit "generalized" and refer to the system of equations derived in [48,49] as first-order hydrodynamics, 1 st GHD, to emphasize that it is a system of first-order partial differential equations.Within 1 st GHD, it was possible to compute the profiles of local observables [48,49,[51][52][53][54][55][56][57], to conjecture an expression for the time evolution of the entanglement entropy [58], and to efficiently calculate Drude weights [59][60][61][62][63]. There are however fundamental questions that can not be addressed within 1 st GHD; diffusive transport [64][65][66][67][68][69] and large-tim...

High-temperature spin dynamics in the Heisenberg chain: Magnon propagation and emerging Kardar-Parisi-Zhang scaling in the zero-magnetization limit

“…More specifically, the broadening of the ballistic front is given by a sub-diffusive power law t 1/3 . Recently, attempts have been made to interprete these findings in the context of GHD [11,27]. We would like to stress that the exponents arising here, while similar to the KPZ exponents, are believed to have a different origin as the dynamical exponent z discussed in section III B 1 below.…”

“…Then, the ballistic front exhibits a subdiffusive scaling originating from the free quasi-particle dispersion, as is well understood for non-interacting lattice models (see Ref. [11] and references therein). Away from the nearly polarized limit, the quasi-particle velocities are renormalized.…”

“…In fact, fitting the profiles to the function F (y) can be used as a method to extract the velocities as long as h 1, as discussed below. The question of whether the t 1/3 scaling survives in the presence of interactions has been discussed in recent works [11,27]. In Ref.…”

Locally quasi-stationary states in noninteracting spin chains

“…The uncertainty of what should be captured by generalised hydrodynamics has however made problematic even the determination of the next orders of perturbation theory in noninteracting spin chains without U (1) symmetry [83].…”

“…Specifically, the attempt of Ref. [83] to develop a higher-order GHD in noninteracting spin chains was unsuccessful, notwithstanding the author made use of the exact solution to the dynamics (which, incidentally, will be proven in this paper). We will show that such attempt has been spoiled by an incorrect definition of space dependent root densities, a concept that was introduced in GHD without a proper definition [35,36,69].…”