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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

1c expansion of the density matrix for the δ -function Bose gas

Michio Jimbo, Tetsuji Miwa

Published Dec 15, 1981• 31 citations
Publisher

Abstract

We study the strong-coupling ($c\ensuremath{\rightarrow}\ensuremath{\infty}$) expansion of the correlation functions for the one-dimensional $\ensuremath{\delta}$-function Bose gas at zero temperature. Making use of the Bethe's-hypothesis wave functions, we calculate exactly the two-point function (up to the order $\frac{1}{{c}^{2}}$) and the $n$-point functions (up to the order $\frac{1}{c}$), both in the finite box and in the thermodynamic limit. This generalizes the recent result of Creamer, Thacker, and Wilkinson for the $\frac{1}{c}$ correction to the two-point function.

DOI
10.1103/physrevd.24.3169

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Cambridge University Press eBooks•Aug 19, 1993•1 citation
We study the strong-coupling (c\ensuremath→\ensuremath∞c\ensuremath{\rightarrow}\ensuremath{\infty}c\ensuremath→\ensuremath∞) expansion of the correlation functions for the one-dimensional \ensuremathδ\ensuremath{\delta}\ensuremathδ-function Bose gas at zero temperature. Making use of the Bethe's-hypothesis wave functions, we calculate exactly the two-point function (up to the order 1c2\frac{1}{{c}^{2}}c21​) and the nnn-point functions (up to the order 1c\frac{1}{c}c1​), both in the finite box and in the thermodynamic limit. This generalizes the recent result of Creamer, Thacker, and Wilkinson for the 1c\frac{1}{c}c1​ correction to the two-point function.
The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.
An intensive study for both the weak coupling and strong coupling limits of the ground state properties of this classic system is presented. Detailed results for specific values of finite NNN are given and from them results for general NNN are determined. We focus on the density matrix and concomitantly its Fourier transform, the occupation numbers, along with the pair correlation function and concomitantly its Fourier transform, the structure factor. These are the signature quantities of the Bose gas. One specific result is that for weak coupling a rational polynomial structure holds despite the transcendental nature of the Bethe equations. All these results are predicated on the Bethe ansatz and are built upon the seminal works of the past.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb–Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).