Lieb–Liniger model

Lieb–Liniger model
Name	Lieb–Liniger model
Field	Theoretical physics
Introduced	1963
Creators	Elliott Lieb; Werner Liniger
Equations	Quantum many-body Schrödinger equation; Bethe ansatz equations

Lieb–Liniger model The Lieb–Liniger model is an exactly solvable one-dimensional quantum many-body model of N bosons with delta-function interactions on a line or ring introduced by Elliott Lieb and Werner Liniger. It provides a paradigmatic link between rigorous results in Mathematical physics, solvable models in Statistical mechanics, and experiments in Atomic physics, Condensed matter physics, and Optical physics. The model underpins theoretical understanding relevant to research at institutions such as Princeton University, Cornell University, Harvard University, University of Cambridge, and Massachusetts Institute of Technology where many subsequent developments originated.

Introduction

The model describes N identical bosons with contact coupling constant c in one spatial dimension, governed by the many-body Schrödinger operator first solved in the seminal paper by Elliott Lieb and Werner Liniger published in 1963. It sits among a class of integrable models alongside the Heisenberg model, the Hubbard model, and the Sinh-Gordon model, and is closely related to the continuum limit of lattice systems studied by researchers affiliated with Max Planck Institute for Physics, Institut Henri Poincaré, and Steklov Institute of Mathematics. Its exact solvability via factorized scattering and coordinate Bethe ansatz made it central to developments by figures such as Hans Bethe, Ludwig Faddeev, Vladimir Korepin, Barry McCoy, and groups at Landau Institute and St. Petersburg State University.

Exact Solution and Bethe Ansatz

Lieb and Liniger solved the model by applying the coordinate Bethe ansatz, producing quantization conditions akin to those in work by Hans Bethe on the Heisenberg spin chain and later formalized in algebraic frameworks by Ludwig Faddeev and Evgeny Sklyanin. The Bethe equations determine rapidities subject to periodic boundary conditions, connecting to the Yang–Baxter relation studied by C.N. Yang and R.J. Baxter. Exact eigenstates and eigenvalues enable computation of the ground state energy and excitation spectrum, with mathematical techniques developed further by Vladimir Korepin, Neil Bogoliubov, Alexander Izergin, and analysts from Cambridge University and University of Oxford. Rigorous spectral analysis has been pursued by scholars associated with Princeton University, IHES, and CNRS.

Physical Properties and Correlation Functions

The model yields exact expressions and asymptotics for observables such as the ground-state energy density, sound velocity, and local pair correlations g2, studied experimentally by groups at MIT, University of Vienna, University of Bonn, and University of Innsbruck. Calculation of dynamical correlation functions invokes form factor expansions and methods pioneered by Vladimir Korepin and collaborators, with connections to the works of Alexander Zamolodchikov and Al.B. Zamolodchikov on integrable quantum field theory. Finite-temperature thermodynamics follows from the thermodynamic Bethe ansatz formulated by C.N. Yang and Chen Ning Yang’s collaborators and analysts from Statistical Laboratory, Cambridge, while long-distance asymptotics relate to conformal field theory techniques developed by John Cardy and Alexander Belavin.

Limiting Cases and Connections (Tonks–Girardeau, Gross–Pitaevskii)

In the impenetrable limit c → ∞ the model maps to the Tonks–Girardeau gas originally discussed by Marvin Girardeau, connecting to free fermions and to studies at University of California, Berkeley and Yale University on fermionization. In the weakly interacting regime c → 0 the mean-field Gross–Pitaevskii description applies, linking to pioneering contributions by Lev Pitaevskii, Eugene Gross, and experimental programs at JILA and NIST. Crossovers between regimes have been analyzed by theorists at École Normale Supérieure, SÉRPA groups, and teams collaborating with European Laboratory for Non-Linear Spectroscopy.

Experimental Realizations and Observations

Ultracold-atom experiments realized the Lieb–Liniger regime using tight transverse confinement and tunable interactions via Feshbach resonance techniques pioneered by laboratories at Colorado State University, Max Planck Institute for Quantum Optics, and Rice University. Key experimental milestones were achieved by groups led by Markus Greiner, Immanuel Bloch, Till Hänsch, and Rudolf Grimm demonstrating correlation functions, momentum distributions, and dynamical behavior predicted by the model. Observations of quantum Newton’s cradle and nonthermal steady states involved collaborations and institutions including Institut d'Optique, Max Planck Institute for the Science of Light, and Australian National University.

Mathematical Extensions and Generalizations

Mathematicians and physicists extended the model to multicomponent systems, spinor gases, and to include internal degrees of freedom via the Yang–Gaudin model studied by M. Gaudin and connections to the Kondo model and nested Bethe ansatz developed by researchers at Rutgers University and University of Tokyo. Quantum integrability techniques from Ludwig Faddeev’s school, algebraic Bethe ansatz, quantum inverse scattering method, and vertex models by Rodney Baxter have been adapted to study boundary conditions, impurity problems, and generalized hydrodynamics as advanced by teams at Princeton Center for Theoretical Science, Cambridge Centre for Theoretical Cosmology, and Perimeter Institute for Theoretical Physics. Contemporary work links to topics pursued by Max Planck Institute for the Physics of Complex Systems, Simons Foundation collaborations, and groups at University of Illinois Urbana–Champaign and University of California, Santa Barbara exploring nonequilibrium dynamics, entanglement entropy, and quantum information aspects.

Category:Quantum many-body physics