Solvable Hydrodynamics of Quantum Integrable Systems

Bulchandani, Vir B.; Vasseur, Romain; Karrasch, Christoph; Moore, Joel E.

doi:10.1103/physrevlett.119.220604

Cited by 195 publications

(221 citation statements)

References 79 publications

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“…In this paper we have studied the fundamentals of generalised hydrodynamics in noninteracting spin chains. GHD was originally developed in a perturbative framework, as the asymptotic solution of time evolution in the limit of large time [35,36] or low inhomogeneity [56,57] in integrable systems. Ref.…”

Section: Resultsmentioning

confidence: 99%

Locally quasi-stationary states in noninteracting spin chains

Fagotti

2020

SciPost Phys.

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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.

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Section: Resultsmentioning

confidence: 99%

Locally quasi-stationary states in noninteracting spin chains

Fagotti

2020

SciPost Phys.

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“…Furthermore, we improve the numerical method proposed in Ref. [55] to solve GHD equations, promoting it from a first order to a second order algorithm in the time step, providing a great stability enhancement.…”

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Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions

2019

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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...

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“…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-time corrections [20][21][22][23] are two of them.…”

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

2017

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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...

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Solvable Hydrodynamics of Quantum Integrable Systems

Cited by 195 publications

References 79 publications

Locally quasi-stationary states in noninteracting spin chains

Locally quasi-stationary states in noninteracting spin chains

Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions

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

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