Locally quasi-stationary states in noninteracting spin chains

Abstract

Locally quasi-stationary states (LQSS) were introduced as inhomogeneous generalisations of stationary states in integrable systems. Roughly speaking, LQSSs look like stationary states, but only locally. Despite their key role in hydrodynamic descriptions, an unambiguous definition of LQSSs was not given. By solving the dynamics in inhomogeneous noninteracting spin chains, we identify the set of LQSSs as a subspace that is invariant under time evolution, and we explicitly construct the latter in a generalised XY model. As a by-product, we exhibit an exact generalised hydrodynamic theory (including "quantum corrections"). Contents4 The two-temperature scenario revisited 18 5 Continuum scaling limit 19 6 Conclusion 22 A Non-equilibrium dynamics in inhomogeneous systems 24 B Weak inhomogeneous limit: an asymptotic expansion 26 B.1 Locally quasi-stationary states 28 B.2 Off-diagonal states 29 References 30 1 A spin operator O ℓ is quasi-localised around a given site ℓ if there is a sequence of connected subsystems Sn ⊂ Sn+1 (Sn is a proper subset of Sn+1), centred around ℓ, such that O ℓ − tr Sn [O ℓ ] tr Sn [I] ⊗ I Sn < e −α|Sn| , with α > 0 and |Sn| the extent of Sn. A quasilocal charge is an additive charge with quasi-localised density -see Ref. [84] for a review on quasilocal charges in integrable spin chains.

Finite-temperature transport in one-dimensional quantum lattice models

“…Note that it is always possible to add a "derivative term" ∝ o x − o x−1 to a charge density (where o x is a local operator), without modifying the total charge. This introduces an ambiguity in the definition of charge densities beyond the leading order [see (Fagotti, 2020) for more details]. In particular, the kernel D k,k (λ, µ) depends on the specific choice of the densities of charges while the Onsager matrix is invariant (De Nardis et al, 2019a).…”

“…Initially, however, its validity could only be established numerically Ilievski and De Nardis, 2017a) or for some special currents Urichuk et al, 2019). The numerical accuracy of ( 108) and its model-independent form triggered a fervent activity aimed at proving it rigorously (Borsi et al, 2020;Fagotti, 2017;Vu and Yoshimura, 2019;Yoshimura and Spohn, 2020) for all Bethe-ansatz integrable models. This endeavour has been concluded by , who reports a complete proof of (108) in the framework of the quantum-inverse scattering method.…”

“…Currently, however, a systematic method to find all sub-leading corrections to Eq. (107) has been devised only in the noninteracting case (Fagotti, 2017(Fagotti, , 2020). Another active research strand is to extend Eq.…”

On the hydrodynamics of unstable excitations

“…The resulting mathematical procedure is based on the solution of a set of coupled nonlinear integral equations, usually carried out numerically, whose sole inputs are: the one-particle eigenvalues of all conserved charges involved in the GGEs characterizing the original left and right systems, the two-particle scattering matrix of the QFT, and the (stable) particle spectrum of the original theory. Since the original proposals [12,13] a plethora of generalizations have been developed, such as the inclusion of force terms [29][30][31], diffusive and higher corrections [17,[32][33][34], noise [35], integrability breaking terms [36][37][38], and much more. There is now even experimental evidence that GHD provides a better description of transport in an atom chip than conventional hydrodynamics [39].…”

Quasi-Stationary States in Ionic Liquid-Liquid Crystal Mixtures at the Nematic-Isotropic Phase Transition

“…Internal longrange forces can bring in additional spatiotemporal correlations to such open systems, giving rise to states with a very long lifetime, the so-called "quasistationary states" (QSS) [10][11][12]. Quasistationary states involving spin chains [13], lattices with infinity of absorbing states [14], and in hydrodynamics on a torus [15] have proved the general character of these states.…”