Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2004.03802 · Physical review. E

Collision rate ansatz for the classical Toda lattice

Herbert E. Spohn

Published Jun 15, 2020• 44 citations
arXivHTMLPDF

Abstract

We consider a generalized Gibbs ensemble of the classical Toda lattice. We establish that the collision rate ansatz follows because (i) the charge-current susceptibility matrix is symmetric and (ii) the stretch current is proportional to the momentum, hence conserved. The method applies also to other integrable many-body systems, either classical or quantum, provided there is a self-conserved current.

DOI
10.1103/physreve.101.060103

Related papers

Collision rate ansatz for quantum integrable systems

Takato Yoshimura, Herbert E. Spohn

For quantum integrable systems the currents averaged with respect to a generalized Gibbs ensemble are revisited. An exact formula is known, which we call ``collision rate ansatz". While there is considerable work to confirm this ansatz in various models, our approach uses the symmetry of the current-charge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a self-conserved current, i.e. some current which is itself conserved. The collision rate ansatz is then derived. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. It is also explained how from the existence of a boost operator a self-conserved current can be deduced.

arXiv:2004.07113•SciPost Physics•Sep 15, 2020•50 citations

Ballistic space-time correlators of the classical toda lattice

Herbert Spohn

Abstract The Toda lattice is an integrable system and its natural space-time stationary states are the generalized Gibbs ensembles (GGE). Of particular physical interest are then the space-time correlations of the conserved fields. To leading order they scale ballistically. We report on the exact solution of the respective generalized hydrodynamic equations linearized around a GGE as background state. Thereby we obtain a concise formula for the family of scaling functions.

arXiv:1911.10825•Journal of Physics A Mathematical and Theoretical•May 11, 2020•23 citations

Generalized hydrodynamics of the classical Toda system

Benjamin Doyon

We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorized scattering of Toda particles, we obtain the generalized free energy and exact average currents and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and, in particular, the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorized scattering of Toda particles.

arXiv:1902.07624•Journal of Mathematical Physics•Jul 1, 2019•104 citations

Breakdown of the Generalized Gibbs Ensemble for Current-Generating Quenches

Marcin Mierzejewski, Peter Prelovšek, Tomaž Prosen

We establish a relation between two hallmarks of integrable systems: the relaxation towards the generalized Gibbs ensemble (GGE) and the dissipationless charge transport. We show that the former one is possible only if the so-called Mazur bound on the charge stiffness is saturated by local conserved quantities. As an example we show how a non-GGE steady state with a current can be generated in the one-dimensional model of interacting spinless fermions with a flux quench. Moreover, an extended GGE involving the quasilocal conserved quantities can be formulated for this case.

arXiv:1405.2557•Physical Review Letters•Jul 11, 2014•62 citations

Generalized Gibbs Ensemble and string-charge relations in nested Bethe Ansatz

Gyorgy Fehér, Balázs Pozsgay

The non-equilibrium steady states of integrable models are believed to be described by the Generalized Gibbs Ensemble (GGE), which involves all local and quasi-local conserved charges of the model. In this work we investigate integrable lattice models solvable by the nested Bethe Ansatz, with group symmetry SU(N) S U ( N ) , N\ge 3 N ≥ 3 . In these models the Bethe Ansatz involves various types of Bethe rapidities corresponding to the “nesting” procedure, describing the internal degrees of freedom for the excitations. We show that a complete set of charges for the GGE can be obtained from the known fusion hierarchy of transfer matrices. The resulting charges are quasi-local in a certain regime in rapidity space, and they completely fix the rapidity distributions of each string type from each nesting level.

arXiv:1909.04470•SciPost Physics•Mar 3, 2020•10 citations

Bloch theorem in the presence of an additional conserved charge

Haruki Watanabe

The Bloch theorem is a general theorem restricting the persistent current associated with a conserved U(1) charge in a ground state or in a thermal equilibrium. It gives an upper bound of the magnitude of the current density, which is inversely proportional to the system size. In a recent paper, Else and Senthil applied the argument for the Bloch theorem to a generalized Gibbs ensemble, assuming the presence of an additional conserved charge, and predicted a nonzero current density in the nonthermal steady state [Else and Senthil, Phys. Rev. B 104, 205132 (2021)]. In this paper, we provide a complementary derivation based on the canonical ensemble, given that the additional charge is strictly conserved within the system by itself. Furthermore, using the example where the additional conserved charge is the momentum operator, we discuss that the persistent current tends to vanish when the system is in contact with an external momentum reservoir in the comoving frame of the reservoir.

arXiv:2107.05211•Physical Review Research•Jan 21, 2022•6 citations
We consider a generalized Gibbs ensemble of the classical Toda lattice. We establish that the collision rate ansatz follows because (i) the charge-current susceptibility matrix is symmetric and (ii) the stretch current is proportional to the momentum, hence conserved. The method applies also to other integrable many-body systems, either classical or quantum, provided there is a self-conserved current.
For quantum integrable systems the currents averaged with respect to a generalized Gibbs ensemble are revisited. An exact formula is known, which we call ``collision rate ansatz". While there is considerable work to confirm this ansatz in various models, our approach uses the symmetry of the current-charge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a self-conserved current, i.e. some current which is itself conserved. The collision rate ansatz is then derived. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. It is also explained how from the existence of a boost operator a self-conserved current can be deduced.
We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorized scattering of Toda particles, we obtain the generalized free energy and exact average currents and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and, in particular, the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorized scattering of Toda particles.
We establish a relation between two hallmarks of integrable systems: the relaxation towards the generalized Gibbs ensemble (GGE) and the dissipationless charge transport. We show that the former one is possible only if the so-called Mazur bound on the charge stiffness is saturated by local conserved quantities. As an example we show how a non-GGE steady state with a current can be generated in the one-dimensional model of interacting spinless fermions with a flux quench. Moreover, an extended GGE involving the quasilocal conserved quantities can be formulated for this case.
The non-equilibrium steady states of integrable models are believed to be described by the Generalized Gibbs Ensemble (GGE), which involves all local and quasi-local conserved charges of the model. In this work we investigate integrable lattice models solvable by the nested Bethe Ansatz, with group symmetry SU(N) S U ( N ) , N\ge 3 N ≥ 3 . In these models the Bethe Ansatz involves various types of Bethe rapidities corresponding to the “nesting” procedure, describing the internal degrees of freedom for the excitations. We show that a complete set of charges for the GGE can be obtained from the known fusion hierarchy of transfer matrices. The resulting charges are quasi-local in a certain regime in rapidity space, and they completely fix the rapidity distributions of each string type from each nesting level.
The Bloch theorem is a general theorem restricting the persistent current associated with a conserved U(1) charge in a ground state or in a thermal equilibrium. It gives an upper bound of the magnitude of the current density, which is inversely proportional to the system size. In a recent paper, Else and Senthil applied the argument for the Bloch theorem to a generalized Gibbs ensemble, assuming the presence of an additional conserved charge, and predicted a nonzero current density in the nonthermal steady state [Else and Senthil, Phys. Rev. B 104, 205132 (2021)]. In this paper, we provide a complementary derivation based on the canonical ensemble, given that the additional charge is strictly conserved within the system by itself. Furthermore, using the example where the additional conserved charge is the momentum operator, we discuss that the persistent current tends to vanish when the system is in contact with an external momentum reservoir in the comoving frame of the reservoir.