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arXiv:math-ph/0106015 · Reviews in Mathematical Physics

GROUND STATE PROPERTIES OF THE NELSON HAMILTONIAN: A GIBBS MEASURE-BASED APPROACH

Volker Betz, Fumio Hiroshima, József Lőrinczi, R. A. Minlos, Herbert E. Spohn

Published Feb 1, 2002• 55 citations
arXivHTMLPDF

Abstract

The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.

DOI
10.1142/s0129055x02001119

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The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.
We present a simulation study of the properties of the Tonks-Girardeau gas, a system of NNN hard-core bosons confined in a one-dimensional harmonic trap. For this system the ground-state wave function is exactly known based on a Bose-Fermi mapping theorem. We employ Nelsons' interpretation of quantum mechanics in which this NNN-body wave function gives rise to a set of stochastic differential equations for the positions of the NNN particles, which can be simulated by standard methods. In particular, real space densities and momentum distributions can simply be determined by time averages along particle trajectories. Employing this approach we are able to significantly extend the range of particle numbers NNN treated numerically compared to earlier approaches, while reproducing all exactly known results for this model. We also show that for the bosons in a harmonic trap, contrary to what has been assumed so far, the momentum distribution reflects the system size scaling of the occupation numbers of the natural orbitals; that is, it can be used to decide on the presence of a Bose-Einstein condensate.
We consider the time evolution of the renormalized Nelson model, which describes N bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles N\gg 1 with coupling constant proportional to N^{-1/2} . First, we show that initial states exhibiting Bose–Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger–Klein–Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger–Klein–Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.
The excitation spectrum of the interacting Bose gas has two branches. The first branch was dominated in the low momentum area and high density of the ground state particles and the second one was dominated in the high momentum area and was existed in the density of the ground state particles which lower than the existed density of the ground state particles of the first branch. The functional integrals method was used. The Hamiltonian was introduced in form of the combination of field operators which was divided into two parts; ground state particles and exicted particles were described respectively the action and partition function were calculated and were introduced partition function poles which were defined as excitation spectrum. According to the definition of excitation energy of ground was equal to zero, parameters were calculated and devided both forms of Bose excitation spectrums in phase space. The interaction between boson particles in the system was replaced by our ansatz interaction. Its behaviour was like the Lennard-Jones potential. The ansatz interaction was mentioned in configuration space was varied with [e -alphar/(betar)2 - e -alphar/(betar)] by assigning α, β as unity with the inversion unit of length.
A solvable model of a generic trapped bosonic mixture, N 1 bosons of mass m 1 and N 2 bosons of mass m 2 trapped in an harmonic potential of frequency ω and interacting by harmonic inter-particle interactions of strengths λ 1 , λ 2 , and λ 12 , is discussed. It has recently been shown for the ground state [J. Phys. A 50 , 295002 (2017)] that in the infinite-particle limit, when the interaction parameters λ 1 ( N 1 − 1), λ 2 ( N 2 − 1), λ 12 N 1 , λ 12 N 2 are held fixed, each of the species is 100% condensed and its density per particle as well as the total energy per particle are given by the solution of the coupled Gross-Pitaevskii equations of the mixture. In the present work we investigate properties of the trapped generic mixture at the infinite-particle limit, and find differences between the many-body and mean-field descriptions of the mixture, despite each species being 100%. We compute analytically and analyze, both for the mixture and for each species, the center-of-mass position and momentum variances, their uncertainty product, the angular-momentum variance, as well as the overlap of the exact and Gross-Pitaevskii wavefunctions of the mixture. The results obtained in this work can be considered as a step forward in characterizing how important are many-body effects in a fully condensed trapped bosonic mixture at the infinite-particle limit.