Euler-scale dynamical correlations in integrable systems with fluid motion

Abstract

We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.

Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems

“…For the classical hard-rod gas considered in Ref. [37], instead, fluid-cell averaging is not necessary and the Euler-scale predictions for the two-point function are simply obtained by averaging in space.…”

“…Only very recently, in Ref. [37], ∆ ( y,0)→(x,t) and the formula in Eq. ( 29) have been numerically evaluated for various interacting integrable models (Lieb-Liniger, sinh-Gordon and the classical hard-rod gas) in different non-equilibrium protocols involving inhomogeneous initial states.…”

“…Moreover, in Ref. [37], the first numerical demonstration of the validity of the Eulerscale formulas in Eq. (29) has been given by comparing them with Monte Carlo simulations of the microscopic hard-rod dynamics.…”

“…[9,10], and later in Refs. [14,15,[17][18][19][20][21][22][23][24][25], has been to the study of ballistic transport from the partitioning protocol initial inhomogeneous state, but the versatility of GHD allows to study various inhomogeneous setups such as the effect of confining potentials [26,27], bump-release protocols [28,29], correlation functions [30][31][32] and entanglement spreading [33][34][35][36][37]. Remarkably, the formalism can also be used to study nonintegrable systems, provided the integrability breaking is weak enough so that on large enough length and time scales, the conserved charges may be assumed not to be broken [26,[38][39][40][41][42][43][44][45][46][47].…”

“…[30] and later numerically evaluated in Ref. [37], which constitutes a central building block of the results presented in Sec. 4.…”