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arXiv:2212.07480 · Physical Review A

Method of difference-differential equations for some Bethe-ansatz-solvable models

Zoran Ristivojević

Published Dec 26, 2022• 5 citations
arXivHTMLPDF

Abstract

In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations, concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, several examples are considered. First, we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the $N$-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.

DOI
10.1103/physreva.106.062216

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The main objective of this thesis is to explain, from the quantum-mechanical point of view, the nature of dark solitons in one-dimensional cold-atom systems. Models of bosons and fermions with contact interactions on a ring are exactly solvable via the Bethe ansatz, and support so-called type-II elementary excitations. These have long been associated with dark solitons of the Gross-Pitaevskii equation due to the similarity of the dispersion relation, despite the completely different physical properties of the states. Fully understanding this connection is our primary aim. We begin by reviewing the Gross-Pitaevskii equation and its dark soliton solutions. Next, we solve the mean-field problem of two coupled one-dimensional Bose-Einstein condensates, with special emphasis on Josephson vortices and their dispersion relation. Predictions are given for possible experimental detection. Then we give a derivation that justifies a method for the extraction of the so-called missing particle number from the dispersion relation of solitonic excitations. A derivation of the finite Bethe ansatz equations for the Lieb-Liniger and Yang-Gaudin models follows. These describe a single species of bosons and two component fermions, respectively. We review the elementary excitations of the Lieb-Linger model, and carry out a comprehensive study of the (much richer) excitations of the Yang-Gaudin model. The thermodynamic limit Bethe ansatz equations for all states of interest in both models are derived, and the 𝘮𝘪𝘴𝘴𝘪𝘯𝘨 𝘱𝘢𝘳𝘵𝘪𝘤𝘭𝘦 𝘯𝘶𝘮𝘣𝘦𝘳 and the closely-related 𝘱𝘩𝘢𝘴𝘦-𝘴𝘵𝘦𝘱 are extracted from the dispersion relations. Next, we develop a method for approximating the finite-system dispersion relation of solitonic excitations from the thermodynamic limit results. Finally, we show that the single particle density and phase profiles of appropriately formed superpositions of type-II states with different momenta exhibit solitonic features. Through this idea, the missing particle number and phase step extracted from the dispersion relation gain physical meaning. Moreover, we use a convolution model to extract the fundamental quantum dark soliton length scale across the range of interactions and momenta. The insight gained in the bosonic case is used to make inferences about dark solitons in the fermionic case. Furthermore, we study the Hess-Fairbank effect in the repulsive Yang-Gaudin model and the fermionic super Tonks-Girardeau regime.

•Jan 1, 2017•0 citations
In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations, concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, several examples are considered. First, we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the NNN-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.
We study a class of Galilean-invariant one-dimensional Bethe ansatz solvable models in the thermodynamic limit. Their rapidity distribution obeys an integral equation with a difference kernel over a finite interval, which does not admit a closed-form solution. We develop a general formalism enabling one to study the moments of the rapidity distribution, showing that they satisfy a difference-differential equation. The derived equation is explicitly analyzed in the case of the Lieb-Liniger model and the moments are analytically calculated. In addition, we obtained the exact information about the ground-state energy at weak repulsion. The obtained results directly enter a number of physically relevant quantities.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
Abstract The Lieb–Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.
The bare existence of this book is due to the amazing fact that the solution of the stationary Schrödinger equation for the one-dimensional Hubbard model can be reduced to a set of algebraic equations, which is tractable in the thermodynamic limit. These equations will be derived in this chapter. We will call them the Lieb-Wu equations to honour E. H. Lieb and F. Y. Wu, who first obtained them [298]. The derivation is based on a method, called the nested (coordinate) Bethe ansatz, which goes back to the seminal articles of C. N. Yang [493] and M. Gaudin [154], who generalized earlier work [60, 296] on exactly solvable models to models with internal degrees of freedom. The roots of the Lieb-Wu equations parameterize the eigenvalues and the eigenstates of the Hamiltonian of the one-dimensional Hubbard model. They encode the complete information about the model. These roots are not explicitly known in the general N -particle case. In the thermodynamic limit ( N →∞), however, only the distributions of the roots in the complex plane matter, and many physical quantities can be calculated as solutions of linear or non-linear Fredholm type integral equations. Moreover, it is sometimes possible to use the Lieb-Wu equations in an implicit way even for finite N , e.g., in the proof of the SO(4) highest weight properties of the eigenstates in Sections 3.D and 3.F of the appendix or in the calculation of their norm in Section 3.5.
We have explained the theory of exactly solvable models on both the quantum and classical levels. We hope that the reader appreciates the beauty and perfection of the Bethe Ansatz. The theory of completely integrable nonlinear partial differential equations plays an equally important role in the book. It is deeply related to the Bethe Ansatz. We hope that our book can be used for further development of the theory of exactly solvable models (we think that the interested reader can find problems to work out in our book). Our book does not close the theory of the Bethe Ansatz; it opens new possibilities for further development. We have explained how to construct eigenfunctions of the Hamiltonian and how to describe the ground state and its excitations. We have also evaluated dispersion curves and scattering matrices for the excitations. The thermodynamics of exactly solvable models is explained in this book in all the details, including the evaluation of temperature correlation functions (even if they depend on real time). The problem of the evaluation of correlation functions (for Bethe Ansatz solvable models) is solved in our book, using the example of the one-dimensional Bose gas (quantum nonlinear Schrödinger equation). It was done using the ideas of the inverse scattering method (the Lax representation, the Riemann-Hilbert problem, the Gel'fand-Levitan-Marchenko equation, etc.). We would like also to emphasize that all other Bethe Ansatz solvable models are constructed in a similar way.
The main objective of this thesis is to explain, from the quantum-mechanical point of view, the nature of dark solitons in one-dimensional cold-atom systems. Models of bosons and fermions with contact interactions on a ring are exactly solvable via the Bethe ansatz, and support so-called type-II elementary excitations. These have long been associated with dark solitons of the Gross-Pitaevskii equation due to the similarity of the dispersion relation, despite the completely different physical properties of the states. Fully understanding this connection is our primary aim. We begin by reviewing the Gross-Pitaevskii equation and its dark soliton solutions. Next, we solve the mean-field problem of two coupled one-dimensional Bose-Einstein condensates, with special emphasis on Josephson vortices and their dispersion relation. Predictions are given for possible experimental detection. Then we give a derivation that justifies a method for the extraction of the so-called missing particle number from the dispersion relation of solitonic excitations. A derivation of the finite Bethe ansatz equations for the Lieb-Liniger and Yang-Gaudin models follows. These describe a single species of bosons and two component fermions, respectively. We review the elementary excitations of the Lieb-Linger model, and carry out a comprehensive study of the (much richer) excitations of the Yang-Gaudin model. The thermodynamic limit Bethe ansatz equations for all states of interest in both models are derived, and the 𝘮𝘪𝘴𝘴𝘪𝘯𝘨 𝘱𝘢𝘳𝘵𝘪𝘤𝘭𝘦 𝘯𝘶𝘮𝘣𝘦𝘳 and the closely-related 𝘱𝘩𝘢𝘴𝘦-𝘴𝘵𝘦𝘱 are extracted from the dispersion relations. Next, we develop a method for approximating the finite-system dispersion relation of solitonic excitations from the thermodynamic limit results. Finally, we show that the single particle density and phase profiles of appropriately formed superpositions of type-II states with different momenta exhibit solitonic features. Through this idea, the missing particle number and phase step extracted from the dispersion relation gain physical meaning. Moreover, we use a convolution model to extract the fundamental quantum dark soliton length scale across the range of interactions and momenta. The insight gained in the bosonic case is used to make inferences about dark solitons in the fermionic case. Furthermore, we study the Hess-Fairbank effect in the repulsive Yang-Gaudin model and the fermionic super Tonks-Girardeau regime.