Quantum generalized hydrodynamics (QGHD) reconstructs the leading quantum corrections to generalized hydrodynamics (GHD), the exact Euler-scale hydrodynamics of one-dimensional integrable quantum fluids: non-zero equal-time correlations between fluid cells, entanglement growth, and quantum fluctuation statistics in genuinely nonequilibrium states. As introduced in 2020 it is a quadratic theory of linear sound waves about a GHD solution, valid only for zero-entropy (split-Fermi-sea) macrostates, and it does not incorporate the diffusive and higher-gradient corrections known for classical GHD. The open problem is the completion of this theory: (a) the higher-order, higher-derivative terms that would constitute the out-of-equilibrium, spatially inhomogeneous, multi-component generalization of nonlinear Luttinger-liquid theory and nonlinear bosonization; (b) the extension to finite-entropy (thermal or generalized-Gibbs) macrostates, where the smoothed Fermi contour invalidates the sharp chiral (one-direction-propagating) modes on which the construction rests; and (c) the combination of quantum fluctuations with diffusive corrections at the next order in space-time scaling.

Background

Many one-dimensional quantum many-body systems are integrable: they possess an infinite family of local conserved quantities. Examples are the Lieb-Liniger gas of bosons with delta-function repulsion and the spin-1/2 XXZ spin chain (a one-dimensional chain of spins coupled by anisotropic nearest-neighbor exchange interactions). Because of these conservation laws such systems do not thermalize; their stationary states are generalized Gibbs ensembles (GGE), statistical ensembles weighted by every conserved charge rather than by energy alone. Their exact large-scale dynamics is generalized hydrodynamics (GHD), formulated in 2016 by Castro-Alvaredo, Doyon and Yoshimura and by Bertini, Collura, De Nardis and Fagotti. The central object of GHD is the local occupation function of Bethe quasiparticles, labeled by a rapidity (the Bethe-ansatz parametrization of quasiparticle momentum), evolved by a kinetic equation whose effective velocities and scattering kernels are fixed by the thermodynamic Bethe ansatz (TBA). GHD is a classical, Euler-scale theory, exact in the simultaneous limit of large distances and long times at fixed ratio; like any Euler-scale hydrodynamics it contains no diffusion, and it predicts vanishing equal-time correlations between distinct fluid cells, so quantum fluctuations are entirely absent from it.

Quantum GHD (QGHD) was introduced by Ruggiero, Calabrese, Doyon and Dubail [Phys. Rev. Lett. 124, 140603 (2020)] to reconstruct the leading quantum corrections to GHD. A Luttinger liquid is the universal quadratic theory of density fluctuations of a one-dimensional quantum fluid with a single pair of Fermi points; a zero-entropy or split-Fermi-sea macrostate has several Fermi points per position, and Eliens and Caux had constructed the corresponding homogeneous, time-independent multi-component Luttinger-liquid theory for such states. QGHD generalizes this to genuinely out-of-equilibrium, spatially inhomogeneous situations: the Fermi contour (the set of Fermi rapidities as a function of position and time) is promoted to a quantum fluctuating object, and one obtains a time-dependent, spatially inhomogeneous, multi-component Luttinger-liquid Hamiltonian whose parameters are fixed by TBA data evaluated along the classical GHD solution. The theory reproduces time-dependent density-matrix renormalization group (tDMRG) benchmarks for the equal-time density-density correlations after a trap quench (a sudden change of the confining potential) of the Lieb-Liniger gas from a double well to a harmonic well, a protocol close to the quantum Newton's cradle experiments on nearly integrable one-dimensional ultracold gases, and was later verified in the Tonks-Girardeau (hard-core) limit, where it also yields exact results for density fluctuations and entanglement entropy.

The construction carries three structural restrictions, all acknowledged in the source Letter. First, it is quadratic: it quantizes linear sound waves propagating on top of a GHD solution, so intrinsically nonlinear quantum phenomena are outside its reach. The Letter notes that higher-order and higher-derivative terms would lead to a generalization of the nonlinear Luttinger liquid (the equilibrium theory accounting for dispersion curvature beyond the harmonic Luttinger-liquid approximation) and of nonlinear bosonization (the reformulation of one-dimensional quantum-fluid dynamics as classical Hamiltonian evolution of the Fermi contour), and that these terms are beyond its scope; its Supplemental Material gives the multi-species generalization of the quadratic theory, but no nonquadratic scheme exists. Second, it requires zero-entropy macrostates: the construction needs sharp Fermi points with well-defined chiral (one-direction-propagating) excitations around them, whereas thermal or GGE macrostates have a smoothed Fermi contour. Third, it is an Euler-scale theory: the subleading diffusive and higher-gradient corrections that are known for classical GHD are not combined with the quantum fluctuation theory. In addition, the theory makes a sharp structural prediction, divergences of the density-density correlator, in the thermodynamic limit, at spacetime points where the number of Fermi points changes, and the quantitative smoothing of these singularities by finite-size, finite-time or resolution effects was left unanalyzed.

Why the restrictions matter: zero-entropy macrostates are nongeneric in experiments (any finite temperature smooths the Fermi contour), nonlinear regimes such as shocks and wave breaking are ubiquitous in expansion dynamics, and observable quantum correlations at late times are shaped by diffusion. A completion of QGHD would be the analogue, for the emergent hydrodynamics of integrable quantum fluids, of what nonlinear Luttinger-liquid theory and nonlinear bosonization are for equilibrium one-dimensional quantum liquids, and would supply the quantum sector that classical fluctuating hydrodynamics of GHD lacks.

Problem Statement

Construct the theory of quantum fluctuations around generalized hydrodynamics beyond the leading quadratic, zero-entropy, Euler-scale approximation, keeping the structure that makes the existing theory exact: an effective theory whose parameters are fixed by thermodynamic Bethe ansatz data along the GHD solution and whose correlation functions remain computable, at least numerically. The single objective is the general quantum fluctuation theory of integrable one-dimensional fluids in generic nonequilibrium situations; the existing 2020 construction is its leading approximation, and its completion requires lifting the three restrictions under which it was derived. The three items below constitute this program, and completing it requires all three:

  1. Nonlinear order. Derive the higher-order and higher-derivative corrections to the quadratic QGHD Hamiltonian (higher powers of the fluctuating Fermi-contour displacement and its derivatives), sufficient to capture intrinsically nonlinear quantum phenomena: the out-of-equilibrium, time-dependent, spatially inhomogeneous, multi-component generalization of nonlinear Luttinger-liquid theory and of nonlinear bosonization, valid for interacting multi-species models, not only the single-species Lieb-Liniger gas.

  2. Finite entropy. Extend the construction to finite-entropy macrostates (thermal or generalized Gibbs ensembles), where the Fermi contour is smoothed and the chiral excitations about sharp Fermi points, on which the zero-entropy construction rests, must be redefined; characterize the interplay between thermal and quantum fluctuations in this regime.

  3. Beyond the Euler scale. Combine the quantum fluctuation theory with the subleading hydrodynamic corrections of GHD, diffusion and higher-gradient terms, yielding a quantum fluctuating hydrodynamics at the next order in space-time scaling (the quantum analogue of the role thermal noise plays in classical fluctuating hydrodynamics, in a setting where classical noise is known to vanish).

These three components are facets of one theory: each removes a different restriction (order of the fluctuation expansion, class of macrostates, space-time scale) from the same construction. A solution must deliver explicit correlation functions beyond the leading approximation; as consistency tests, it must reduce to the current theory and to the equilibrium multi-component Luttinger liquid of Eliens and Caux in the appropriate limits, and it must reproduce the benchmarks already established for the current theory, namely the equal-time density-density correlations after the trap quench and, in the Tonks-Girardeau limit, the density fluctuations and entanglement entropy.

References

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