Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions

Abstract

We provide a new hydrodynamic framework to describe out-of-equilibrium integrable systems with space-time inhomogeneous interactions. Our result builds up on the recently-introduced Generalized Hydrodynamics (GHD). The method allows to analytically describe the dynamics during generic space-time-dependent smooth modulations of the interactions. As a proof of concept, we study experimentally-motivated interaction quenches in the trapped interacting Bose gas, which cannot be treated with current analytical or numerical methods. We also benchmark our results in the XXZ spin chain and in the classical Sinh-Gordon model.Introduction. -Exploring the out-of-equilibrium behavior of quantum many-body systems is nowadays among the most active research areas in physics, due to a successful synergy between theoretical and experimental advances [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].How, and in which sense, does a coarse-grained thermodynamic description emerge through dynamical evolution in isolated out-of-equilibrium many-body systems? One-dimensional systems represent an ideal playground to address this question: there, remarkably powerful tools exist, both theoretical (such as conformal field theory [18] and integrability [19,20]) and computational (such as Matrix Product States methods [21]).Integrability is ubiquitous in the low-dimensional world, with applications ranging from spin chains [19] to continuum models (having Lorentz [20] or Galilean [22,23] invariance, or neither [24]). Amazingly, many of these examples have been experimentally realized [9][10][11][12][13][14][15][16][17].Integrable models are characterized by the presence of infinitely many conserved chargesQ j , which can be used to exactly determine their thermodynamics [25]. In recent times, the importance of quasi-local charges has moreover been underlined [26][27][28][29][30][31][32][33][34][35].The last decade has witnessed exact results reaching out-of-equilibrium protocols as well: great attention has been devoted to the homogeneous sudden quantum quench [36] (see also Ref. [37] and reference therein). Due to the conserved quantities, the system exhibits local relaxation to a state that is not thermal [38][39][40][41][42][43][44][45][46], but rather emerges from a Quench Action [47,48] or (where applicable) a Generalized Gibbs Ensemble [49,50] which accounts for all the relevant charges.More recently, the focus has been on quenches from spatially inhomogeneous systems. A new theoretical toolbox, dubbed Generalized Hydrodynamics (GHD) [51,52] allows to address this problem. In Ref. [51,52] GHD dealt with inhomogeneous states evolving under a homogeneous Hamiltonian. Several applications have been explored , extending the initial findings to describe the entanglement spreading [79][80][81][82][83][84], including diffusive corrections [85][86][87][88] or applying it to classical field theories [89][90][91][92]. Very recently, it has been shown that GHD FIG. 1: Prototypical experimental setup that can be addressed with our method...

Locally quasi-stationary states in noninteracting spin chains

“…In addition, the theory is unable to capture diffusive behaviour in interacting systems, a subject that has been attracting more and more attention [66,67,70,71,[78][79][80]. In order to overcame this weakness, there have been proposals to go beyond the lowest order (of an implicit asymptotic expansion in the limit of low inhomogeneity) in interacting integrable systems [78,79,81,82], but, at the same time, some issues in noninteracting spin chains were uncovered. Specifically, the attempt of Ref.…”

Introducing iFluid: a numerical framework for solving hydrodynamical equations in integrable models

“…Furthermore, space-time inhomogeneities in the parameters of the model induce force terms on the quasiparticles, which can change their rapidities. Once again the interparticle interactions collectively dress these force terms, whereby the quasiparticles experience an effective acceleration [27] a eff (λ) =…”

“…Finally, we are ready to solve the GHD equation (9), using the SecondOrderSolver class [27]. Simply pass the model to the solver and run the simulation through the propagateTheta() method: 25…”

“…Thus, an infinite set of advection equations emerge, which through the thermodynamic Bethe ansatz, can be formulated as a single Euler-scale equation for a quasiparticle distribution. Since the inception of GHD several applications have been added to the framework, such as calculations of entanglement spreading [16][17][18][19], correlation functions [20][21][22], diffusive corrections [23,24], and many others [25][26][27][28]. Recently it has also been demonstrated to capture the dynamics of a cold Bose gas trapped on an atom-chip [29].…”

Entanglement spreading and quasiparticle picture beyond the pair structure

“…[5] and reference therein). Very recently, weakly inhomogeneous and timedependent protocols have been made accessible through the Generalized Hydrodynamics [29][30][31][32]. Furthermore, spreading of both correlations and entanglement has been framed in a simple, yet powerful, quasi-particle picture [33].…”