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:cond-mat/0504041 · Physical Review Letters

Quantum Hydrodynamics, the Quantum Benjamin-Ono Equation, and the Calogero Model

Alexander G. Abanov, Paul B. Wiegmann

Published Aug 11, 2005• 104 citations
arXivHTMLPDF

Abstract

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.

DOI
10.1103/physrevlett.95.076402

Related papers

Quantum dynamics of the effective field theory of the Calogero-Sutherland model

Federico L. Bottesi, Guillermo R. Zemba

We consider the known effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles, obtained from the standard procedure in conformal field theory: the Hilbert space is constructed a priori in terms of irreducible representations of the symmetry algebra, and not by diagonalization of the hamiltonian, which is given in terms of fields that carry representations of the W 1 + ∞ algebra (representing the incompressibility of the Fermi sea). Nevertheless, the role of the effective hamiltonian of the theory is to establish a specific dynamics, which deserves further consideration. We show that the time evolution of the (chiral or antichiral) density field is given by the quantum Benjamin-Ono equation, in agreement with previous results obtained from the alternative description of the continuous limit of the model, based on quantum hydrodynamics. In this study, all calculations are performed at the quantum operator level, without making any assumption on the semiclassical limit of the fields and their equations of motion. This result may be considered as a reliable indication of the equivalence between the quantum field theoretic and quantum hydrodynamical formulations of the effective theories of the model. A one-dimensional quantum compressible fluid that includes both chiralities is the physical picture that emerges for the continuous limit of the Calogero-Sutherland model. • The chiral density field of the Calogero-Sutherland EFT satisfies a quantum Benjamin-Ono equation. • Agrees with quantum hydrodynamics results by Abanov and Wiegmann, extending their validity to quantum fields themselves. • Findings could be of interest to the QFT community, with new tools to deal with time non-locality within the realm of CFT. • Equivalence between the EFT and the hydrodynamic formulations of the thermodynamic limit of the Calogero-Sutherland model is supported.

arXiv:2404.10011•Nuclear Physics B•Feb 7, 2025•2 citations

A hydrodynamical description of Bose liquid with fractional exclusion statistics

Zhaoyu Fei, Yu Chen

Abstract Hydrodynamical systems are usually taken as chaotic systems with fast relaxations. It is counter intuitive for ‘ideal’ gas to have a hydrodynamical description. We find that a hydrodynamical model of one-dimensional | Φ | 6 theory shares the same ground state density profile, density-wave excitation, as well as the similar dynamical and statistical properties with the Calogero–Sutherland model in thermodynamic limit when their interaction strengths matches each other. The interaction strength g 0 in | Φ | 6 theory is then the index of fractional statistics. Although the model is interacting in Bose liquid sense, but it shows integrability with periodical coherent evolution. We also discussed the fractional statistics emerges from the | Φ | 6 theory.

arXiv:2306.06853•New Journal of Physics•Feb 1, 2024•2 citations

Fine structure of the phonon in one dimension from quantum hydrodynamics

Tom Price, Austen Lamacraft

We show that the resonant interactions between phonons in one dimension may be treated consistently within quantum hydrodynamics by the introduction of phonon dispersion. In this way the physics of a nonlinear Luttinger liquid may be described in terms of hydrodynamic (i.e., bosonized) variables without the introduction of impurities at the outset, and gives a complementary view on the mobile impurity model from the hydrodynamics. We focus on the calculation of the dynamic structure factor for a model with quadratic dispersion, which has the Benjamin-Ono equation of fluid dynamics as its equation of motion. We find singular behavior in the vicinity of upper and lower energetic thresholds corresponding to phonon and soliton branches of the classical theory, which may be benchmarked against known results for the Calogero-Sutherland model.

arXiv:1405.7372•Physical Review B•Dec 19, 2014•11 citations

Integrable quantum hydrodynamics in two-dimensional phase space

E Bettelheim

Quantum liquids in two dimensions represent interesting dynamical quantum systems for several reasons, among them the possibility of the existence of infinite hidden symmetries, such as conformal symmetry or the symmetry associated with area preserving diffeomorphisms. It is known that when the symmetry algebra is large enough, symmetry may fully prescribe the dynamics. However, the way this is borne out in two-dimensional hydrodynamics, both classical and quantum, is not fully understood. Here we take a step towards clarifying this issue, by focusing on a particular example, namely that of a two-dimensional phase space liquid which emerges when one considers the Calogero model, a many-body one-dimensional system interacting through an inverse square law potential. We demonstrate how the symmetry algebra of conserved quantities of the one-dimensional system is expressed in terms of the incompressible Euler hydrodynamics of point vortices in phase space. Due to a formal relation between quantum hydrodynamics and classical stochastic hydrodynamics, which is inherent in the method of stochastic quantization, the ideas and methods developed here may also have application in the study of stochastic classical hydrodynamics, after suitable modifications.

arXiv:1306.3782•Journal of Physics A Mathematical and Theoretical•Nov 26, 2013•2 citations

Modified kinetic theory of Bose systems taking into account slow hydrodynamical processes

Petro A. Hlushak, М. В. Токарчук

An approach to the description of kinetics which taking into account the large-scale hydrodynamic transport processes for quantum Bose system is proposed. The nonequilibrium statistical operator which consistently describes both the kinetic and nonlinear hydrodynamic fluctuations in quantum liquid is calculated. Using this operator the coupled equations for quantum one-particle distribution function and functional of hydrodynamic variables: densities of momentum, energies and number of particles are obtained.

arXiv:1112.5849•Dec 26, 2011•0 citations

Laminar flow of charged quantum fluids of the Calogero–Sutherland universality class

Federico L. Bottesi, Guillermo R. Zemba

The effective field theory of the Calogero–Sutherland model represents a universality class of quantum hydrodynamic fluids in one spatial dimension. It describes quantum compressible fluids involving both chiralities in which the chiral density field obeys the quantum Benjamin–Ono equation. An extension of this theory to describe a laminar flow of the Calogero–Sutherland fluids in a rectangular geometry with small transverse width and the topology of a ribbon, is considered here. The physical picture is based on the edge states in the hierarchical quantum Hall effect, which may be seen as a collection of parallel one-dimensional quantum incompressible fluids moving along but confined within the transverse microscopic width of the edge of the sample. The effective theory is thus defined as the direct product of two one-dimensional theories of the Calogero–Sutherland class so that one involves motion while the other is confining. Charge transport may be induced by coupling the system to an external electromagnetic field that yields a global translation of the ground state. The effective theory describes quantum solitonic excitations along the direction of the flow and possesses a two-dimensional electric current density which shows a Wigner semicircle law profile in the transverse direction, suggesting a Poiseuille-like behavior but without dissipative viscous effects since the velocity of the fluid is not a well-defined quantum field. This simple physical picture predicts interesting phenomena with distinctive signatures that may be tested in real samples.

arXiv:2305.06744•International Journal of Modern Physics A•Nov 8, 2024•2 citations
Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.
We consider the known effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles, obtained from the standard procedure in conformal field theory: the Hilbert space is constructed a priori in terms of irreducible representations of the symmetry algebra, and not by diagonalization of the hamiltonian, which is given in terms of fields that carry representations of the W 1 + ∞ algebra (representing the incompressibility of the Fermi sea). Nevertheless, the role of the effective hamiltonian of the theory is to establish a specific dynamics, which deserves further consideration. We show that the time evolution of the (chiral or antichiral) density field is given by the quantum Benjamin-Ono equation, in agreement with previous results obtained from the alternative description of the continuous limit of the model, based on quantum hydrodynamics. In this study, all calculations are performed at the quantum operator level, without making any assumption on the semiclassical limit of the fields and their equations of motion. This result may be considered as a reliable indication of the equivalence between the quantum field theoretic and quantum hydrodynamical formulations of the effective theories of the model. A one-dimensional quantum compressible fluid that includes both chiralities is the physical picture that emerges for the continuous limit of the Calogero-Sutherland model. • The chiral density field of the Calogero-Sutherland EFT satisfies a quantum Benjamin-Ono equation. • Agrees with quantum hydrodynamics results by Abanov and Wiegmann, extending their validity to quantum fields themselves. • Findings could be of interest to the QFT community, with new tools to deal with time non-locality within the realm of CFT. • Equivalence between the EFT and the hydrodynamic formulations of the thermodynamic limit of the Calogero-Sutherland model is supported.
Abstract Hydrodynamical systems are usually taken as chaotic systems with fast relaxations. It is counter intuitive for ‘ideal’ gas to have a hydrodynamical description. We find that a hydrodynamical model of one-dimensional | Φ | 6 theory shares the same ground state density profile, density-wave excitation, as well as the similar dynamical and statistical properties with the Calogero–Sutherland model in thermodynamic limit when their interaction strengths matches each other. The interaction strength g 0 in | Φ | 6 theory is then the index of fractional statistics. Although the model is interacting in Bose liquid sense, but it shows integrability with periodical coherent evolution. We also discussed the fractional statistics emerges from the | Φ | 6 theory.
We show that the resonant interactions between phonons in one dimension may be treated consistently within quantum hydrodynamics by the introduction of phonon dispersion. In this way the physics of a nonlinear Luttinger liquid may be described in terms of hydrodynamic (i.e., bosonized) variables without the introduction of impurities at the outset, and gives a complementary view on the mobile impurity model from the hydrodynamics. We focus on the calculation of the dynamic structure factor for a model with quadratic dispersion, which has the Benjamin-Ono equation of fluid dynamics as its equation of motion. We find singular behavior in the vicinity of upper and lower energetic thresholds corresponding to phonon and soliton branches of the classical theory, which may be benchmarked against known results for the Calogero-Sutherland model.
Quantum liquids in two dimensions represent interesting dynamical quantum systems for several reasons, among them the possibility of the existence of infinite hidden symmetries, such as conformal symmetry or the symmetry associated with area preserving diffeomorphisms. It is known that when the symmetry algebra is large enough, symmetry may fully prescribe the dynamics. However, the way this is borne out in two-dimensional hydrodynamics, both classical and quantum, is not fully understood. Here we take a step towards clarifying this issue, by focusing on a particular example, namely that of a two-dimensional phase space liquid which emerges when one considers the Calogero model, a many-body one-dimensional system interacting through an inverse square law potential. We demonstrate how the symmetry algebra of conserved quantities of the one-dimensional system is expressed in terms of the incompressible Euler hydrodynamics of point vortices in phase space. Due to a formal relation between quantum hydrodynamics and classical stochastic hydrodynamics, which is inherent in the method of stochastic quantization, the ideas and methods developed here may also have application in the study of stochastic classical hydrodynamics, after suitable modifications.
An approach to the description of kinetics which taking into account the large-scale hydrodynamic transport processes for quantum Bose system is proposed. The nonequilibrium statistical operator which consistently describes both the kinetic and nonlinear hydrodynamic fluctuations in quantum liquid is calculated. Using this operator the coupled equations for quantum one-particle distribution function and functional of hydrodynamic variables: densities of momentum, energies and number of particles are obtained.
The effective field theory of the Calogero–Sutherland model represents a universality class of quantum hydrodynamic fluids in one spatial dimension. It describes quantum compressible fluids involving both chiralities in which the chiral density field obeys the quantum Benjamin–Ono equation. An extension of this theory to describe a laminar flow of the Calogero–Sutherland fluids in a rectangular geometry with small transverse width and the topology of a ribbon, is considered here. The physical picture is based on the edge states in the hierarchical quantum Hall effect, which may be seen as a collection of parallel one-dimensional quantum incompressible fluids moving along but confined within the transverse microscopic width of the edge of the sample. The effective theory is thus defined as the direct product of two one-dimensional theories of the Calogero–Sutherland class so that one involves motion while the other is confining. Charge transport may be induced by coupling the system to an external electromagnetic field that yields a global translation of the ground state. The effective theory describes quantum solitonic excitations along the direction of the flow and possesses a two-dimensional electric current density which shows a Wigner semicircle law profile in the transverse direction, suggesting a Poiseuille-like behavior but without dissipative viscous effects since the velocity of the fluid is not a well-defined quantum field. This simple physical picture predicts interesting phenomena with distinctive signatures that may be tested in real samples.