Exact Euler-scale multi-point correlations in interacting integrable fluids out of equilibrium, and the singular regimes of the effective velocity

Generalized hydrodynamics (GHD) gives the exact Euler-scale dynamics of one-dimensional integrable many-body systems out of equilibrium. Doyon's 2018 framework derived exact Euler-scale two-point correlation functions in inhomogeneous, non-stationary states of interacting integrable fluids, formulated a recursive procedure for n-point functions, and solved all n-point functions in free models; it explicitly left open the evaluation of n-point ($n\geq 3$) functions in interacting fluids, and observed that all established asymptotic formulae degenerate at spectral points where the effective velocity is stationary in the rapidity, with modified decay naively expected as $t^{-1/2}$ instead of $1/t$. Later work settled parts of this agenda: stationary-state three-point Euler amplitudes now follow from the nonlinear projection formula (Doyon 2025), diffusive corrections to two-point functions are understood in shock-free systems, and classical-model validations exist. What remains open is the completion of the theory in generic nonequilibrium situations: explicit, evaluable Euler-scale $n\geq 3$ point correlation functions in interacting inhomogeneous, non-stationary fluids (including the long-range correlations that such states develop), and the exact correlation behaviour at stationary points of the effective velocity where the established formulae break down.

Background

One-dimensional integrable many-body systems - such as the Lieb-Liniger Bose gas, the XXZ spin chain, the Hubbard model, and relativistic quantum field theories like the sinh-Gordon model - admit an extensive set of local conservation laws. Their exact thermodynamics is described by the thermodynamic Bethe ansatz (TBA), which represents the states relevant to local observations as fluids of quasiparticles labelled by a rapidity $\theta$ (a momentum-like parameter) whose only interaction is a two-body scattering phase. A generalized Gibbs ensemble (GGE) is the maximal-entropy state fixed by the occupation function $n(\theta)$ of these quasiparticles. Generalized hydrodynamics (GHD), discovered in 2016, is the exact large-scale hydrodynamics of such systems: the evolution of $n(x,t,\theta)$ is a kinetic equation in which the quasiparticle velocity is the effective velocity $v^{\rm eff}(\theta)$, obtained from the bare velocity by 'dressing', a linear integral equation involving the scattering kernel and the occupation function that accounts for the position shifts quasiparticles accumulate in collisions. The GHD equation is solvable by characteristics: the trajectory of a quasiparticle of rapidity $\theta$ through a slowly varying fluid is determined implicitly by the initial profile $n_0$ and the dressing equations.

Correlations at the Euler scale are defined by simultaneously scaling space and time ($x\to\lambda x$, $t\to\lambda t$, $\lambda\to\infty$): connected n-point functions of local observables decay as $\lambda^{1-n}$, which is the least-decaying, 'ballistic' part of the full correlation functions. Doyon (2018) combined an Euler-scale fluctuation-dissipation principle with the solution by characteristics to derive, for inhomogeneous and non-stationary GGE-type initial states of interacting integrable fluids: (i) exact two-point functions of conserved densities and currents, expressed through a propagator - the linear response of the time-evolved occupation function to localized perturbations of the conjugate fields, fixed by a linear integral equation along the characteristics; (ii) two-point functions of arbitrary local fields as Leclair-Mussardo-type infinite form-factor series, in which matrix elements between particle-hole excitations over the TBA fluid replace the vacuum form factors of the usual form-factor expansion; (iii) a recursive procedure able, in principle, to generate n-point functions through higher functional derivatives of the evolution map $n_0\mapsto n_t$; and (iv) all Euler-scale n-point functions in free models. In the partitioning protocol (two semi-infinite halves joined at the origin), the connected two-point function along a ray decays at large time generically as $1/t$, with explicit factors $1/|\partial_\theta v^{\rm eff}|$ evaluated at the rapidity whose effective velocity equals the ray velocity.

The paper left the core of this programme open. First, the development of expressions for higher-point functions in interacting fluids was deferred: the propagator integral equations and the recursion were formulated, but the higher functional derivatives of $n_0\mapsto n_t$ were never evaluated for $n\geq 3$. Second, the paper observed that all asymptotic formulae break down at spectral points where $\partial_\theta v^{\rm eff}(\theta)=0$ while the quasiparticle density at that rapidity is nonzero: there the effective velocity is stationary in the rapidity, quasiparticles accumulate, and the decay law is expected to change, naively from $1/t$ to $t^{-1/2}$, with analogous singularities in fully inhomogeneous situations where the rapidity derivative of the characteristic function vanishes.

The subsequent literature clarified the structure of both questions without closing them. Nonlinear response theory (Fava et al. 2021) showed that higher-point functions in stationary states involve nonlinear hydrodynamic projections, but under assumptions that were later found incomplete: ballistic macroscopic fluctuation theory (Doyon, Perfetto, Sasamoto, Yoshimura 2023) showed that spatial long-range correlations generically emerge in nonequilibrium states and invalidate naive local entropy maximisation at higher orders. Doyon (2025) then derived the exact nonlinear projection formula - a sum over 'minimal connected covers' expressing connected n-point functions of arbitrary observables through those of conserved densities, valid for all $n$ and all spatial dimensions - and evaluated it to obtain explicit three-point Euler amplitudes in homogeneous stationary states, while stating that higher-point functions in stationary states and long-range correlations in non-stationary states remain to be analysed. The question of what happens where the effective velocity is stationary has not been touched at all. The open problem is therefore to complete the exact Euler-scale correlation theory of interacting integrable fluids in generic out-of-equilibrium situations: beyond two-point functions, and beyond the non-degenerate velocity regimes that all established formulae assume.

Problem Statement

Complete the exact Euler-scale correlation theory of interacting integrable fluids in generic out-of-equilibrium situations, by solving the two linked problems that the source framework (Doyon, SciPost Phys. 5, 054 (2018)) explicitly left open.

(1) Higher-point functions in inhomogeneous, non-stationary interacting fluids. Derive and explicitly evaluate the connected Euler-scale n-point ($n\geq 3$) correlation functions of local observables in interacting integrable models (quantum Bethe-ansatz-solvable systems or classical integrable gases) evolving from inhomogeneous, non-stationary GGE-type initial states. This means implementing the source's recursive procedure - the higher functional derivatives of the evolution map $n_0\mapsto n_t$ defined by the GHD characteristics and the propagator integral equations - including the effect of the hydrodynamic long-range correlations that non-stationary states develop, and producing closed-form or algorithmically evaluable expressions in TBA data (occupation function, dressing equations, effective velocities, static covariances). At minimum, a complete answer must: (i) give general expressions for $n=3$ valid at arbitrary space-time points in inhomogeneous, non-stationary backgrounds, together with a well-defined algorithm specifying the extension to arbitrary $n$; (ii) reduce, in the free-quasiparticle limit, to the known free-model Euler-scale n-point functions; (iii) reduce, in homogeneous stationary states, to the published three-point Euler amplitudes, and for $n=2$ to the source's propagator formulae.

(2) Singular regimes of the effective velocity. Determine the behaviour of the Euler-scale correlation functions at and near spectral points where the effective velocity is stationary in the rapidity, $\partial_\theta v^{\rm eff}(\theta)=0$, with nonzero quasiparticle density there: establish the modified large-time decay along ballistic rays (the source's naive expectation is a $t^{-1/2}$ modification of the generic $1/t$ decay, to be confirmed, refuted, or refined), the correlation structure generated by the accumulation of quasiparticles at stationary effective velocity including light-cone-boundary effects, and the analogous singularities in fully inhomogeneous situations where the rapidity derivative of the characteristic function vanishes.

These two components jointly complete one and the same objective: component (1) extends the exact correlation theory in the number of correlated observables, component (2) extends its domain of validity in rapidity space beyond the non-degenerate regime assumed by every established formula. A solution is complete when the explicit higher-point theory of (1) is available and the decay law and correlation structure of (2) are established, each with the stated reductions and consistency checks.

References

1. B. Doyon, Exact large-scale correlations in integrable systems out of equilibrium, SciPost Phys. 5, 054 (2018), doi:10.21468/SciPostPhys.5.5.054, arXiv:1711.04568, https://arxiv.org/abs/1711.04568
2. O. A. Castro-Alvaredo, B. Doyon, T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016), doi:10.1103/PhysRevX.6.041065, https://doi.org/10.1103/PhysRevX.6.041065
3. B. Bertini, M. Collura, J. De Nardis, M. Fagotti, Transport in out-of-equilibrium XXZ chains: exact profiles of charges and currents, Phys. Rev. Lett. 117, 207201 (2016), doi:10.1103/PhysRevLett.117.207201, https://doi.org/10.1103/PhysRevLett.117.207201
4. B. Doyon, T. Yoshimura, A note on generalized hydrodynamics: inhomogeneous fields and other concepts, SciPost Phys. 2, 014 (2017), doi:10.21468/SciPostPhys.2.2.014, arXiv:1611.08225, https://arxiv.org/abs/1611.08225
5. B. Doyon, H. Spohn, T. Yoshimura, A geometric viewpoint on generalized hydrodynamics, Nucl. Phys. B 926, 570 (2018), doi:10.1016/j.nuclphysb.2017.12.002, arXiv:1704.04409, https://arxiv.org/abs/1704.04409
6. A. Bastianello, B. Doyon, G. Watts, T. Yoshimura, Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model, SciPost Phys. 4, 045 (2018), doi:10.21468/SciPostPhys.4.6.045, arXiv:1712.05687, https://arxiv.org/abs/1712.05687
7. B. Pozsgay, I. M. Szécsényi, LeClair-Mussardo series for two-point functions in integrable QFT, JHEP 2018, 170 (2018), doi:10.1007/JHEP05(2018)170, arXiv:1802.05890, https://arxiv.org/abs/1802.05890
8. J. De Nardis, M. Panfil, Edge singularities and quasi-long-range order in non-equilibrium steady states, Phys. Rev. Lett. 120, 217206 (2018), doi:10.1103/PhysRevLett.120.217206, arXiv:1801.08079, https://arxiv.org/abs/1801.08079
9. A. Bastianello, V. Alba, J.-S. Caux, Generalized hydrodynamics with space-time inhomogeneous interactions, Phys. Rev. Lett. 123, 130602 (2019), doi:10.1103/PhysRevLett.123.130602, arXiv:1906.01654, https://arxiv.org/abs/1906.01654
10. J. De Nardis, D. Bernard, B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6, 049 (2019), doi:10.21468/SciPostPhys.6.4.049, arXiv:1812.00767, https://arxiv.org/abs/1812.00767
11. F. S. Møller, G. Perfetto, B. Doyon, J. Schmiedmayer, Euler-scale dynamical correlations in integrable systems with fluid motion, SciPost Phys. Core 3, 016 (2020), doi:10.21468/SciPostPhysCore.3.2.016, arXiv:2007.00527, https://arxiv.org/abs/2007.00527
12. M. Fava, S. Biswas, G. Gopalakrishnan, R. Vasseur, S. A. Parameswaran, Hydrodynamic nonlinear response of interacting integrable systems, Proc. Natl. Acad. Sci. 118, e2106945118 (2021), doi:10.1073/pnas.2106945118, arXiv:2103.06899, https://arxiv.org/abs/2103.06899
13. G. Perfetto, B. Doyon, Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems, SciPost Phys. 10, 116 (2021), doi:10.21468/SciPostPhys.10.5.116, arXiv:2012.06496, https://arxiv.org/abs/2012.06496
14. A. Hutsalyuk, B. Pozsgay, L. Pristyák, The LeClair-Mussardo series and nested Bethe Ansatz, Nucl. Phys. B 964, 115306 (2021), doi:10.1016/j.nuclphysb.2021.115306, arXiv:2009.13203, https://arxiv.org/abs/2009.13203
15. J. De Nardis, B. Doyon, M. Medenjak, M. Panfil, Correlation functions and transport coefficients in generalised hydrodynamics, J. Stat. Mech. (2022) 014002, doi:10.1088/1742-5468/ac3658, arXiv:2104.04462, https://arxiv.org/abs/2104.04462
16. M. Panfil, R. M. Konik, Extending the thermodynamic form factor bootstrap program: multiple particle-hole excitations, crossing symmetry, and reparameterization invariance, JHEP 2023, 072 (2023), doi:10.1007/JHEP10(2023)072, arXiv:2306.10748, https://arxiv.org/abs/2306.10748
17. B. Doyon, G. Perfetto, T. Sasamoto, T. Yoshimura, Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems, Phys. Rev. Lett. 131, 027101 (2023), doi:10.1103/PhysRevLett.131.027101, https://doi.org/10.1103/PhysRevLett.131.027101
18. B. Doyon, G. Perfetto, T. Sasamoto, T. Yoshimura, Ballistic macroscopic fluctuation theory, SciPost Phys. 15, 136 (2023), doi:10.21468/SciPostPhys.15.4.136, https://doi.org/10.21468/SciPostPhys.15.4.136
19. G. Del Vecchio Del Vecchio, M. Kormos, B. Doyon, A. Bastianello, Exact large-scale fluctuations of the phase field in the sine-Gordon model, Phys. Rev. Lett. 131, 263401 (2023), doi:10.1103/PhysRevLett.131.263401, arXiv:2305.10495, https://arxiv.org/abs/2305.10495
20. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, R. Vasseur, Generalized hydrodynamics: a perspective, Phys. Rev. X 15, 010501 (2025), doi:10.1103/PhysRevX.15.010501, arXiv:2311.03438, https://arxiv.org/abs/2311.03438
21. F. Hübner, B. Doyon, Existence and uniqueness of solutions to the generalized hydrodynamics equation, arXiv:2411.04922 (2024), https://arxiv.org/abs/2411.04922
22. F. Hübner, L. Biagetti, J. De Nardis, B. Doyon, Diffusive hydrodynamics from long-range correlations, Phys. Rev. Lett. 134, 187101 (2025), doi:10.1103/PhysRevLett.134.187101, arXiv:2408.04502, https://arxiv.org/abs/2408.04502
23. B. Doyon, Nonlinear projection for ballistic correlation functions: a formula in terms of minimal connected covers, arXiv:2506.05266 (2025), https://arxiv.org/abs/2506.05266
24. B. Doyon, Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models, SciPost Phys. 20, 111 (2026), doi:10.21468/SciPostPhys.20.4.111, arXiv:2506.05279, https://arxiv.org/abs/2506.05279
25. E. Charnay, A. Escoubet, F. Copie, S. Randoux, T. Bonnemain, A. Bastianello, P. Suret, Experimental observation of ballistic correlations in integrable turbulence, arXiv:2601.21085 (2026), https://arxiv.org/abs/2601.21085