Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium

Abstract

Understanding the general principles underlying strongly interacting quantum states out of equilibrium is one of the most important tasks of current theoretical physics. With experiments accessing the intricate dynamics of many-body quantum systems, it is paramount to develop powerful methods that encode the emergent physics. Up to now, the strong dichotomy observed between integrable and non-integrable evolutions made an overarching theory difficult to build, especially for transport phenomena where space-time profiles are drastically different. We present a novel framework for studying transport in integrable systems: hydrodynamics with infinitely-many conservation laws. This bridges the conceptual gap between integrable and non-integrable quantum dynamics, and gives powerful tools for accurate studies of space-time profiles. We apply it to the description of energy transport between heat baths, and provide a full description of the current-carrying nonequilibrium steady state and the transition regions in a family of models including the Lieb-Liniger model of interacting Bose gases, realized in experiments. arXiv:1605.07331v3 [cond-mat.stat-mech]

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

“…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].…”

“…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.…”

“…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.…”

“…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.…”

“…The firstorder hydrodynamic equation is recovered in the limit of weak inhomogeneity, which allows one to expand the last two terms of (9) about x, ignoring the contributions from spatial derivatives higher than the first. That is the equation that Refs [48,49] generalized to interacting integrable models. Incidentally, applying it to (9) the same prescription that lifted its first-order approximation to a theory for interacting integrable models results in ∂ t ρ hyd n;x,t + v hyd n;x+ 1 2 ρ hyd n;x+ 1…”