Exact large-scale correlations in integrable systems out of equilibrium
Benjamin Doyon
SciPost Phys. 5, 054 (2018) · published 28 November 2018
- doi: 10.21468/SciPostPhys.5.5.054
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Abstract
Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate $n$-point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyze correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale $n$-point functions.
TY - JOUR
PB - SciPost Foundation
DO - 10.21468/SciPostPhys.5.5.054
TI - Exact large-scale correlations in integrable systems out of equilibrium
PY - 2018/11/28
UR - https://www.scipost.org/SciPostPhys.5.5.054
JF - SciPost Physics
JA - SciPost Phys.
VL - 5
IS - 5
SP - 054
A1 - Doyon, Benjamin
AB - Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate $n$-point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyze correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale $n$-point functions.
ER -@Article{10.21468/SciPostPhys.5.5.054,
title={{Exact large-scale correlations in integrable systems out of equilibrium}},
author={Benjamin Doyon},
journal={SciPost Phys.},
volume={5},
pages={054},
year={2018},
publisher={SciPost},
doi={10.21468/SciPostPhys.5.5.054},
url={https://scipost.org/10.21468/SciPostPhys.5.5.054},
}
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-
Kundu et al., Dynamical correlations of conserved quantities in the one-dimensional equal mass hard particle gas
J. Stat. Mech. 2020, 023205 (2020) [Crossref] -
Kormos et al., Finite-temperature dynamics in gapped one-dimensional models in the sine-Gordon family
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Doyon et al., Ballistic macroscopic fluctuation theory
SciPost Phys. 15, 136 (2023) [Crossref] -
Bonnemain et al., Two-dimensional stationary soliton gas
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Bastianello et al., Landau-Lifschitz Magnets: Exact Thermodynamics and Transport
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Friedman et al., Integrable Many-Body Quantum Floquet-Thouless Pumps
Phys. Rev. Lett. 123, 170603 (2019) [Crossref] -
Mazzuca et al., Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
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Fava et al., Long-time divergences in the nonlinear response of gapped one-dimensional many-particle systems
SciPost Phys. 19, 086 (2025) [Crossref] -
VU et al., Equations of state in generalized hydrodynamics
SciPost Phys. 6, 023 (2019) [Crossref] -
Doyon et al., Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems
Phys. Rev. Lett. 131, 027101 (2023) [Crossref] -
Panfil et al., The relevant excitations for the one-body function in the Lieb–Liniger model
J. Stat. Mech. 2021, 073103 (2021) [Crossref] -
Bastianello et al., Hydrodynamics of weak integrability breaking
J. Stat. Mech. 2021, 114003 (2021) [Crossref] -
Watson et al., Benchmarks of generalized hydrodynamics for one-dimensional Bose gases
Phys. Rev. Research 5, L022024 (2023) [Crossref] -
Doyon, Generalized hydrodynamics of the classical Toda system
60, 073302 (2019) [Crossref] -
Gopalakrishnan et al., Anomalous relaxation and the high-temperature structure factor of XXZ spin chains
Proc. Natl. Acad. Sci. U.S.A. 116, 16250 (2019) [Crossref] -
Yoshimura et al., Collision rate ansatz for quantum integrable systems
SciPost Phys. 9, 040 (2020) [Crossref] -
Perfetto et al., Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems
SciPost Phys. 10, 116 (2021) [Crossref] -
Friedman et al., Diffusive hydrodynamics from integrability breaking
Phys. Rev. B 101, 180302 (2020) [Crossref] -
Hübner et al., Quadrature for the generalized hydrodynamics equation and absence of shocks in the Lieb-Liniger model
Phys. Rev. B 112, L241101 (2025) [Crossref] -
Granet, Low-density limit of dynamical correlations in the Lieb–Liniger model
J. Phys. A: Math. Theor. 54, 154001 (2021) [Crossref] -
Nozawa et al., Generalized hydrodynamics study of the one-dimensional Hubbard model: Stationary clogging and proportionality of spin, charge, and energy currents
Phys. Rev. B 103, 035130 (2021) [Crossref] -
Doyon et al., Generalized Hydrodynamics: A Perspective
Phys. Rev. X 15, 010501 (2025) [Crossref] -
Gawędzki et al., Finite-Time Universality in Nonequilibrium CFT
J Stat Phys 172, 353 (2018) [Crossref] -
Hutsalyuk et al., The LeClair-Mussardo series and nested Bethe Ansatz
Nuclear Physics B 964, 115306 115306 (2021) [Crossref] -
Doyon et al., Fluctuations in Ballistic Transport from Euler Hydrodynamics
Ann. Henri Poincaré 21, 255 (2020) [Crossref] -
De Nardis et al., Correlation functions and transport coefficients in generalised hydrodynamics
J. Stat. Mech. 2022, 014002 (2022) [Crossref] -
Fehér et al., The propagator of the finite XXZ spin-$\tfrac{1}{2}$ chain
SciPost Phys. 6, 063 (2019) [Crossref] -
Gerbino et al., Kinetics of quantum reaction-diffusion systems
SciPost Phys. Core 8, 014 (2025) [Crossref] -
De Nardis et al., Diffusion in generalized hydrodynamics and quasiparticle scattering
SciPost Phys. 6, 049 (2019) [Crossref] -
Mukherjee et al., Hydrodynamics of a hard-core active lattice gas
Phys. Rev. E 111, 024128 (2025) [Crossref] -
Fagotti, Locally quasi-stationary states in noninteracting spin chains
SciPost Phys. 8, 048 (2020) [Crossref] -
Ruggiero et al., Quantum Generalized Hydrodynamics
Phys. Rev. Lett. 124, 140603 (2020) [Crossref] -
Doyon, Lecture notes on Generalised Hydrodynamics
SciPost Phys. Lect. Notes, 18 (2020) [Crossref] -
Mestyán et al., Spin-charge separation effects in the low-temperature transport of one-dimensional Fermi gases
Phys. Rev. B 99, 014305 (2019) [Crossref] -
Alba et al., Generalized-hydrodynamic approach to inhomogeneous quenches: correlations, entanglement and quantum effects
J. Stat. Mech. 2021, 114004 (2021) [Crossref] -
Doyon, Diffusion and Superdiffusion from Hydrodynamic Projections
J Stat Phys 186, 25 (2022) [Crossref] -
Del Vecchio Del Vecchio et al., The hydrodynamic theory of dynamical correlation functions in the XX chain
J. Stat. Mech. 2022, 053102 (2022) [Crossref] -
Alba et al., Entanglement evolution and generalised hydrodynamics: interacting integrable systems
SciPost Phys. 7, 005 (2019) [Crossref] -
Krajnik et al., Integrable matrix models in discrete space-time
SciPost Phys. 9, 038 (2020) [Crossref] -
Bastianello et al., Observation of a generalized Gibbs ensemble in photonics
Phys. Rev. A 113, 013514 (2026) [Crossref] -
Cortes Cubero et al., Generalized hydrodynamics regime from the thermodynamic bootstrap program
SciPost Phys. 8, 004 (2020) [Crossref] -
Essler, A short introduction to Generalized Hydrodynamics
Physica A: Statistical Mechanics and its Applications 631, 127572 127572 (2023) [Crossref] -
Bastianello, Lack of thermalization for integrability-breaking impurities
EPL 125, 20001 (2019) [Crossref] -
Bonnemain et al., Hamiltonian Formulation and Aspects of Integrability of Generalised Hydrodynamics
Ann. Henri Poincaré, (2025) [Crossref] -
Fava et al., Divergent Nonlinear Response from Quasiparticle Interactions
Phys. Rev. Lett. 131, 256505 (2023) [Crossref] -
Agrawal et al., Generalized hydrodynamics, quasiparticle diffusion, and anomalous local relaxation in random integrable spin chains
Phys. Rev. B 99, 174203 (2019) [Crossref] -
Spohn, Ballistic space-time correlators of the classical toda lattice
J. Phys. A: Math. Theor. 53, 265004 (2020) [Crossref] -
Gopalakrishnan et al., Anomalous transport from hot quasiparticles in interacting spin chains
Rep. Prog. Phys. 86, 036502 (2023) [Crossref] -
Doyon, Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems
Commun. Math. Phys. 391, 293 (2022) [Crossref] -
Møller et al., Euler-scale dynamical correlations in integrable systems with fluid motion
SciPost Phys. Core 3, 016 (2020) [Crossref] -
Panfil, The two particle–hole pairs contribution to the dynamic correlation functions of quantum integrable models
J. Stat. Mech. 2021, 013108 (2021) [Crossref] -
Hübner et al., Diffusive Hydrodynamics from Long-Range Correlations
Phys. Rev. Lett. 134, 187101 (2025) [Crossref] -
Gerbino et al., Large-scale universality in quantum reaction-diffusion from Keldysh field theory
Phys. Rev. B 109, L220304 (2024) [Crossref] -
Miao, Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models
SciPost Phys. 11, 066 (2021) [Crossref] -
Granet et al., Systematic strong coupling expansion for out-of-equilibrium dynamics in the Lieb-Liniger model
SciPost Phys. 11, 068 (2021) [Crossref] -
de Klerk et al., Improved Hilbert space exploration algorithms for finite temperature calculations
SciPost Phys. Core 6, 039 (2023) [Crossref] -
Bouchoule et al., Generalized hydrodynamics in the one-dimensional Bose gas: theory and experiments
J. Stat. Mech. 2022, 014003 (2022) [Crossref] -
Panfil et al., Linearized regime of the generalized hydrodynamics with diffusion
SciPost Phys. Core 1, 002 (2019) [Crossref] -
Myers et al., Transport fluctuations in integrable models out of equilibrium
SciPost Phys. 8, 007 (2020) [Crossref] -
Bastianello et al., Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions
Phys. Rev. Lett. 123, 130602 (2019) [Crossref] -
Bonnemain et al., Soliton gas of the integrable Boussinesq equation and its generalised hydrodynamics
SciPost Phys. 18, 075 (2025) [Crossref] -
Perfetto et al., Dynamics of large deviations in the hydrodynamic limit: Noninteracting systems
Phys. Rev. E 102, 042128 (2020) [Crossref] -
Ampelogiannis et al., Long-Time Dynamics in Quantum Spin Lattices: Ergodicity and Hydrodynamic Projections at All Frequencies and Wavelengths
Ann. Henri Poincaré 25, 65 (2024) [Crossref] -
Alba, Towards a generalized hydrodynamics description of Rényi entropies in integrable systems
Phys. Rev. B 99, 045150 (2019) [Crossref] -
Borsi et al., Current operators in integrable models: a review
J. Stat. Mech. 2021, 094001 (2021) [Crossref] -
Robinson et al., On computing non-equilibrium dynamics following a quench
SciPost Phys. 11, 104 (2021) [Crossref] -
Bertini et al., Finite-temperature transport in one-dimensional quantum lattice models
Rev. Mod. Phys. 93, 025003 (2021) [Crossref] -
Mazzuca, Generalized hydrodynamics for the volterra lattice: ballistic and non-ballistic behavior of correlation functions
J. Phys. A: Math. Theor. 57, 395207 (2024) [Crossref] -
Borsi et al., Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence
Phys. Rev. X 10, 011054 (2020) [Crossref] -
De Nardis et al., Anomalous Spin Diffusion in One-Dimensional Antiferromagnets
Phys. Rev. Lett. 123, 186601 (2019) [Crossref] -
Essler et al., Statistics of Matrix Elements of Local Operators in Integrable Models
Phys. Rev. X 14, 031048 (2024) [Crossref] -
Pawełczyk, Thermodynamics and generalized hydrodynamics of simple integrable QFT in finite volume
J. Phys. A: Math. Theor. 57, 495002 (2024) [Crossref] -
Bastianello et al., Integrability-Protected Adiabatic Reversibility in Quantum Spin Chains
Phys. Rev. Lett. 122, 240606 (2019) [Crossref] -
Cubero et al., Form factors and generalized hydrodynamics for integrable systems
J. Stat. Mech. 2021, 114002 (2021) [Crossref] -
Fava et al., Hydrodynamic nonlinear response of interacting integrable systems
Proc. Natl. Acad. Sci. U.S.A. 118, e2106945118 (2021) [Crossref] -
Møller et al., Whitham approach to Generalized Hydrodynamics
Phys. Rev. Research 6, 013328 (2024) [Crossref] -
Nozawa et al., Generalized hydrodynamic approach to charge and energy currents in the one-dimensional Hubbard model
Phys. Rev. B 101, 035121 (2020) [Crossref] -
Bocini, Connected correlations in partitioning protocols: A case study and beyond
SciPost Phys. 15, 027 (2023) [Crossref] -
Rylands et al., Transport and Entanglement across Integrable Impurities from Generalized Hydrodynamics
Phys. Rev. Lett. 131, 156303 (2023) [Crossref] -
Koch et al., Generalized hydrodynamics of the attractive non-linear Schrӧdinger equation
J. Phys. A: Math. Theor. 55, 134001 (2022) [Crossref] -
Wang et al., Emergent ballistic transport of Bose–Fermi mixtures in one dimension
J. Phys. A: Math. Theor. 53, 464002 (2020) [Crossref] -
Panfil et al., Extending the thermodynamic form factor bootstrap program: multiple particle-hole excitations, crossing symmetry, and reparameterization invariance
J. High Energ. Phys. 2023, 72 (2023) [Crossref] -
Bastianello et al., Generalized hydrodynamics with dephasing noise
Phys. Rev. B 102, 161110 (2020) [Crossref] -
Granet et al., Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions
SciPost Phys. 9, 033 (2020) [Crossref] -
Horváth et al., Symmetry resolved entanglement in integrable field theories via form factor bootstrap
J. High Energ. Phys. 2020, 131 (2020) [Crossref] -
Granet et al., Out-of-equilibrium dynamics of the XY spin chain from form factor expansion
SciPost Phys. 12, 019 (2022) [Crossref] -
Vu, Cumulants of conserved charges in GGE and cumulants of total transport in GHD: exact summation of matrix elements?
J. Stat. Mech. 2020, 023103 (2020) [Crossref] -
Møller et al., Introducing iFluid: a numerical framework for solving hydrodynamical equations in integrable models
SciPost Phys. 8, 041 (2020) [Crossref] -
Koch et al., Adiabatic formation of bound states in the one-dimensional Bose gas
Phys. Rev. B 103, 165121 (2021) [Crossref] -
Møller et al., Extension of the Generalized Hydrodynamics to the Dimensional Crossover Regime
Phys. Rev. Lett. 126, 090602 (2021) [Crossref] -
Granet et al., A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model
SciPost Phys. 9, 082 (2020) [Crossref] -
Klobas et al., Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54
SciPost Phys. 11, 106 (2021) [Crossref] -
Bonnemain et al., Generalized hydrodynamics of the KdV soliton gas
J. Phys. A: Math. Theor. 55, 374004 (2022) [Crossref]