SciPost Physics

SciPost Phys. 3, 039 (2017) · published 15 December 2017

• doi: 10.21468/SciPostPhys.3.6.039
• pdf
• BiBTeX
• RIS
• Submissions/Reports
• 

Abstract

Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive Lieb-Liniger $\delta$-Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a by-product we obtain the dynamical two-point correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.

• Bastianello et al., Hydrodynamics of weak integrability breaking
  J. Stat. Mech. 2021, 114003 (2021) [Crossref]
• Krajnik et al., Exact Anomalous Current Fluctuations in a Deterministic Interacting Model
  Phys. Rev. Lett. 128, 160601 (2022) [Crossref]
• Kuniba et al., Current correlations, Drude weights and large deviations in a box–ball system
  J. Phys. A: Math. Theor. 55, 244006 (2022) [Crossref]
• Doyon et al., Ballistic macroscopic fluctuation theory
  SciPost Phys. 15, 136 (2023) [Crossref]
• Manna et al., Thermal Drude weight in an integrable chiral clock model
  Phys. Rev. B 108, 054304 (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]
• De Nardis et al., Correlation functions and transport coefficients in generalised hydrodynamics
  J. Stat. Mech. 2022, 014002 (2022) [Crossref]
• Fagotti, Locally quasi-stationary states in noninteracting spin chains
  SciPost Phys. 8, 048 (2020) [Crossref]
• Friedman et al., Diffusive hydrodynamics from integrability breaking
  Phys. Rev. B 101, 180302 (2020) [Crossref]
• Agrawal et al., Anomalous low-frequency conductivity in easy-plane XXZ spin chains
  Phys. Rev. B 101, 224415 (2020) [Crossref]
• Doyon et al., Fluctuations in Ballistic Transport from Euler Hydrodynamics
  Ann. Henri Poincaré 21, 255 (2020) [Crossref]
• Borsi et al., Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence
  Phys. Rev. X 10, 011054 (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]
• 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]
• Gamayun et al., Finite temperature spin diffusion in the Hubbard model in the strong coupling limit
  SciPost Phys. 15, 073 (2023) [Crossref]
• Bastianello et al., From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics
  J. Stat. Mech. 2018, 113104 (2018) [Crossref]
• Bertini et al., Finite-temperature transport in one-dimensional quantum lattice models
  Rev. Mod. Phys. 93, 025003 (2021) [Crossref]
• De Nardis et al., Hydrodynamic Diffusion in Integrable Systems
  Phys. Rev. Lett. 121, 160603 (2018) [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]
• Cecile et al., Squeezed Ensembles and Anomalous Dynamic Roughening in Interacting Integrable Chains
  Phys. Rev. Lett. 132, 130401 (2024) [Crossref]
• Bertini et al., Universal Broadening of the Light Cone in Low-Temperature Transport
  Phys. Rev. Lett. 120, 176801 (2018) [Crossref]
• Ilievski et al., Superdiffusion in One-Dimensional Quantum Lattice Models
  Phys. Rev. Lett. 121, 230602 (2018) [Crossref]
• Doyon et al., Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems
  Phys. Rev. Lett. 131, 027101 (2023) [Crossref]
• Kundu et al., Dynamical correlations of conserved quantities in the one-dimensional equal mass hard particle gas
  J. Stat. Mech. 2020, 023205 (2020) [Crossref]
• Doyon, Lecture notes on Generalised Hydrodynamics
  SciPost Phys. Lect. Notes, 18 (2020) [Crossref]
• Mazza et al., Energy transport in an integrable parafermionic chain via generalized hydrodynamics
  Phys. Rev. B 98, 075421 (2018) [Crossref]
• Kormos et al., Semiclassical theory of front propagation and front equilibration following an inhomogeneous quantum quench
  Phys. Rev. E 98, 032105 (2018) [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]
• 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]
• Doyon et al., Free energy fluxes and the Kubo–Martin–Schwinger relation
  J. Stat. Mech. 2021, 043206 (2021) [Crossref]
• Bertini et al., Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics
  Phys. Rev. B 100, 035108 (2019) [Crossref]
• Myers et al., Transport fluctuations in integrable models out of equilibrium
  SciPost Phys. 8, 007 (2020) [Crossref]
• Fava et al., Spin crossovers and superdiffusion in the one-dimensional Hubbard model
  Phys. Rev. B 102, 115121 (2020) [Crossref]
• Mastropietro et al., Canonical Drude Weight for Non-integrable Quantum Spin Chains
  J Stat Phys 172, 379 (2018) [Crossref]
• Schemmer et al., Generalized Hydrodynamics on an Atom Chip
  Phys. Rev. Lett. 122, 090601 (2019) [Crossref]
• Krajnik et al., Anisotropic Landau-Lifshitz model in discrete space-time
  SciPost Phys. 11, 051 (2021) [Crossref]
• Bastianello et al., Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model
  SciPost Phys. 4, 045 (2018) [Crossref]
• Bastianello et al., Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions
  Phys. Rev. Lett. 123, 130602 (2019) [Crossref]
• Alba, Towards a generalized hydrodynamics description of Rényi entropies in integrable systems
  Phys. Rev. B 99, 045150 (2019) [Crossref]
• Lopez-Piqueres et al., Integrability breaking in the Rule 54 cellular automaton
  J. Phys. A: Math. Theor. 55, 234005 (2022) [Crossref]
• Bouchoule et al., Generalized hydrodynamics in the one-dimensional Bose gas: theory and experiments
  J. Stat. Mech. 2022, 014003 (2022) [Crossref]
• Collura et al., Analytic solution of the domain-wall nonequilibrium stationary state
  Phys. Rev. B 97, 081111 (2018) [Crossref]
• Ilievski, Popcorn Drude weights from quantum symmetry
  J. Phys. A: Math. Theor. 55, 504005 (2022) [Crossref]
• De Nardis et al., Particle-hole pairs and density–density correlations in the Lieb–Liniger model
  J. Stat. Mech. 2018, 033102 (2018) [Crossref]
• Fava et al., Hydrodynamic nonlinear response of interacting integrable systems
  Proc. Natl. Acad. Sci. U.S.A. 118, e2106945118 (2021) [Crossref]
• Robinson et al., On computing non-equilibrium dynamics following a quench
  SciPost Phys. 11, 104 (2021) [Crossref]
• Ruggiero et al., Conformal field theory on top of a breathing one-dimensional gas of hard core bosons
  SciPost Phys. 6, 051 (2019) [Crossref]
• Durnin et al., Diffusive hydrodynamics of inhomogenous Hamiltonians
  J. Phys. A: Math. Theor. 54, 494001 (2021) [Crossref]
• De Nardis et al., Anomalous Spin Diffusion in One-Dimensional Antiferromagnets
  Phys. Rev. Lett. 123, 186601 (2019) [Crossref]
• Pristyák et al., Current mean values in the XYZ model
  SciPost Phys. 14, 158 (2023) [Crossref]
• Hübner, Mesoscopic impurities in generalized hydrodynamics
  J. Stat. Mech. 2024, 033102 (2024) [Crossref]
• Medenjak et al., TT¯ -Deformed Conformal Field Theories out of Equilibrium
  Phys. Rev. Lett. 126, 121601 (2021) [Crossref]
• Doyon, Generalized hydrodynamics of the classical Toda system
  60, 073302 (2019) [Crossref]
• Bastianello et al., Generalized hydrodynamics with dephasing noise
  Phys. Rev. B 102, 161110 (2020) [Crossref]
• Tanaka et al., Thermal conductivity of a weakly interacting Bose gas in quasi-one-dimension
  Phys. Rev. E 106, 064104 (2022) [Crossref]
• Urichuk et al., Nonlinear Transport by Bethe Bound States
  Phys. Rev. Lett. 129, 096602 (2022) [Crossref]
• Perfetto et al., Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems
  SciPost Phys. 10, 116 (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]
• Spohn, Generalized Gibbs Ensembles of the Classical Toda Chain
  J Stat Phys 180, 4 (2020) [Crossref]
• Bonnemain et al., Generalized hydrodynamics of the KdV soliton gas
  J. Phys. A: Math. Theor. 55, 374004 (2022) [Crossref]
• Hua et al., Dynamics of nonequilibrium magnons in gapped Heisenberg antiferromagnets
  Phys. Rev. B 109, 054306 (2024) [Crossref]
• Ilievski et al., Superuniversality of Superdiffusion
  Phys. Rev. X 11, 031023 (2021) [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]
• Dhar et al., Revisiting the Mazur bound and the Suzuki equality
  Chaos, Solitons & Fractals 144, 110618 110618 (2021) [Crossref]
• Karevski et al., Charge-current correlation equalities for quantum systems far from equilibrium
  SciPost Phys. 6, 068 (2019) [Crossref]
• Krajnik et al., Universal anomalous fluctuations in charged single-file systems
  Phys. Rev. Research 6, 013260 (2024) [Crossref]
• Gawędzki et al., Finite-Time Universality in Nonequilibrium CFT
  J Stat Phys 172, 353 (2018) [Crossref]
• Nagy et al., Thermodynamics and fractal Drude weights in the sine-Gordon model
  Phys. Rev. B 108, L241105 (2023) [Crossref]
• Popkov et al., Quest for the golden ratio universality class
  Phys. Rev. E 109, 044111 (2024) [Crossref]
• Bulchandani et al., Superdiffusion in spin chains
  J. Stat. Mech. 2021, 084001 (2021) [Crossref]
• Yoshimura et al., Collision rate ansatz for quantum integrable systems
  SciPost Phys. 9, 040 (2020) [Crossref]
• Krajnik et al., Undular Diffusion in Nonlinear Sigma Models
  Phys. Rev. Lett. 125, 240607 (2020) [Crossref]
• Mazzuca et al., Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
  J Stat Phys 190, 149 (2023) [Crossref]
• VU et al., Equations of state in generalized hydrodynamics
  SciPost Phys. 6, 023 (2019) [Crossref]
• Doyon, Diffusion and Superdiffusion from Hydrodynamic Projections
  J Stat Phys 186, 25 (2022) [Crossref]
• Alba, Entanglement and quantum transport in integrable systems
  Phys. Rev. B 97, 245135 (2018) [Crossref]
• Friedman et al., Integrable Many-Body Quantum Floquet-Thouless Pumps
  Phys. Rev. Lett. 123, 170603 (2019) [Crossref]
• Doyon, Exact large-scale correlations in integrable systems out of equilibrium
  SciPost Phys. 5, 054 (2018) [Crossref]
• Gopalakrishnan et al., Superdiffusion from Nonabelian Symmetries in Nearly Integrable Systems
  Annu. Rev. Condens. Matter Phys. 15, 159 (2024) [Crossref]
• Urichuk et al., Analytical results for the low-temperature Drude weight of the XXZ spin chain
  Phys. Rev. B 103, 245108 (2021) [Crossref]
• De Nardis et al., Diffusion in generalized hydrodynamics and quasiparticle scattering
  SciPost Phys. 6, 049 (2019) [Crossref]
• Cubero et al., Form factors and generalized hydrodynamics for integrable systems
  J. Stat. Mech. 2021, 114002 (2021) [Crossref]
• Cortes Cubero et al., Generalized hydrodynamics regime from the thermodynamic bootstrap program
  SciPost Phys. 8, 004 (2020) [Crossref]
• Urichuk et al., The spin Drude weight of the XXZ chain and generalized hydrodynamics
  SciPost Phys. 6, 005 (2019) [Crossref]
• Sirker, Transport in one-dimensional integrable quantum systems
  SciPost Phys. Lect. Notes, 17 (2020) [Crossref]
• Bastianello, Lack of thermalization for integrability-breaking impurities
  EPL 125, 20001 (2019) [Crossref]
• Krajnik et al., Integrable matrix models in discrete space-time
  SciPost Phys. 9, 038 (2020) [Crossref]
• De Nardis et al., Superdiffusion from Emergent Classical Solitons in Quantum Spin Chains
  Phys. Rev. Lett. 125, 070601 (2020) [Crossref]
• Medenjak et al., Thermal transport in TT¯ -deformed conformal field theories: From integrability to holography
  Phys. Rev. D 103, 066012 (2021) [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]
• Gopalakrishnan et al., Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains
  Phys. Rev. Lett. 122, 127202 (2019) [Crossref]
• Ruggiero et al., Quantum Generalized Hydrodynamics
  Phys. Rev. Lett. 124, 140603 (2020) [Crossref]
• Jin et al., Interplay between transport and quantum coherences in free fermionic systems
  J. Phys. A: Math. Theor. 54, 404001 (2021) [Crossref]
• De Nardis et al., Stability of Superdiffusion in Nearly Integrable Spin Chains
  Phys. Rev. Lett. 127, 057201 (2021) [Crossref]
• Bastianello et al., Superluminal moving defects in the Ising spin chain
  Phys. Rev. B 98, 064304 (2018) [Crossref]
• Koch et al., Adiabatic formation of bound states in the one-dimensional Bose gas
  Phys. Rev. B 103, 165121 (2021) [Crossref]
• Lopez-Piqueres et al., Hydrodynamics of nonintegrable systems from a relaxation-time approximation
  Phys. Rev. B 103, L060302 (2021) [Crossref]
• Møller et al., Introducing iFluid: a numerical framework for solving hydrodynamical equations in integrable models
  SciPost Phys. 8, 041 (2020) [Crossref]
• De Nardis et al., Universality Classes of Spin Transport in One-Dimensional Isotropic Magnets: The Onset of Logarithmic Anomalies
  Phys. Rev. Lett. 124, 210605 (2020) [Crossref]
• Agrawal et al., Generalized hydrodynamics, quasiparticle diffusion, and anomalous local relaxation in random integrable spin chains
  Phys. Rev. B 99, 174203 (2019) [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]
• Gopalakrishnan et al., Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems
  Phys. Rev. B 98, 220303 (2018) [Crossref]
• Krajnik et al., Absence of Normal Fluctuations in an Integrable Magnet
  Phys. Rev. Lett. 128, 090604 (2022) [Crossref]
• Bastianello et al., Nonequilibrium Steady State Generated by a Moving Defect: The Supersonic Threshold
  Phys. Rev. Lett. 120, 060602 (2018) [Crossref]
• Spohn, Interacting and noninteracting integrable systems
  59, 091402 (2018) [Crossref]
• Bastianello et al., Integrability-Protected Adiabatic Reversibility in Quantum Spin Chains
  Phys. Rev. Lett. 122, 240606 (2019) [Crossref]
• Ae et al., Spin Drude weight for the integrable XXZ chain with arbitrary spin
  J. Stat. Mech. 2024, 033104 (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]
• Medenjak et al., Diffusion from convection
  SciPost Phys. 9, 075 (2020) [Crossref]
• Wang et al., Emergent ballistic transport of Bose–Fermi mixtures in one dimension
  J. Phys. A: Math. Theor. 53, 464002 (2020) [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]
• Cecile et al., Hydrodynamic relaxation of spin helices
  Phys. Rev. B 108, 075135 (2023) [Crossref]
• Mestyán et al., Molecular dynamics simulation of entanglement spreading in generalized hydrodynamics
  SciPost Phys. 8, 055 (2020) [Crossref]
• Bastianello et al., Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas
  Phys. Rev. Lett. 120, 190601 (2018) [Crossref]
• Pavlis et al., Dressed excitations, thermodynamics and relaxation in the XXZ Heisenberg model
  J. Stat. Mech. 2020, 013101 (2020) [Crossref]
• Cao et al., Incomplete Thermalization from Trap-Induced Integrability Breaking: Lessons from Classical Hard Rods
  Phys. Rev. Lett. 120, 164101 (2018) [Crossref]
• Bertini et al., Entanglement evolution and generalised hydrodynamics: noninteracting systems
  J. Phys. A: Math. Theor. 51, 39LT01 (2018) [Crossref]
• Møller et al., Extension of the Generalized Hydrodynamics to the Dimensional Crossover Regime
  Phys. Rev. Lett. 126, 090602 (2021) [Crossref]
• De Nardis et al., Hydrodynamic gauge fixing and higher order hydrodynamic expansion
  J. Phys. A: Math. Theor. 56, 245001 (2023) [Crossref]