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Nuclear Physics B

Effective large-scale model of boson gas from microscopic theory

Juha Honkonen, Marina Vladimirovna Komarova, Yu. G. Molotkov, Mikhail Yu. Nalimov

Published Dec 20, 2018• 16 citations
Publisher

Abstract

An effective large-scale model of interacting boson gas at low temperatures is constructed from first principles. The starting point is the generating function of time-dependent Green functions at finite temperature. The perturbation expansion is worked out for the generic case of finite time interval and grand-canonical density operator with the use of the S-matrix functional for the generating function. Apparent infrared divergences of the perturbation expansion are pointed out. Regularization via attenuation of propagators is proposed and the relation to physical dissipation is studied. Problems of functional-integral representation of Green functions are analyzed. The proposed large-scale model is explicitly renormalized at the leading order.

DOI
10.1016/j.nuclphysb.2018.12.015

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An effective large-scale model of interacting boson gas at low temperatures is constructed from first principles. The starting point is the generating function of time-dependent Green functions at finite temperature. The perturbation expansion is worked out for the generic case of finite time interval and grand-canonical density operator with the use of the S-matrix functional for the generating function. Apparent infrared divergences of the perturbation expansion are pointed out. Regularization via attenuation of propagators is proposed and the relation to physical dissipation is studied. Problems of functional-integral representation of Green functions are analyzed. The proposed large-scale model is explicitly renormalized at the leading order.
We construct a quantum kinetic theory of a weakly interacting critical boson gas using the expectation values of products of Heisenberg field operators in the grand canonical ensemble. Using a functional representation for the Wick theorem for time-ordered products, we construct a perturbation theory for the generating functional of these time-dependent Green’s functions at a finite temperature. We note some problems of the functional-integral representation and discuss unusual apparent divergences of the perturbation expansion. We propose a regularization of these divergences using attenuating propagators. Using a linear transformation to variables with well-defined scaling dimensions, we construct an infrared effective field theory. We show that the structure of the regularized model is restored by renormalization. We propose a multiplicatively renormalizable infrared effective model of the quantum dynamics of a boson gas.
Dynamic behaviour of a boson gas near the condensation transition in the symmetric phase is analyzed with the use of an effective large-scale model derived from time-dependent Green functions at finite temperature. A renormalization-group analysis shows that the scaling exponents of critical dynamics of the effective multi-charge model coincide with those of the standard model A. The departure of this result from the description of the superfluid transition by either model E or F of the standard phenomenological stochastic models is corroborated by the analysis of a generalization of model F, which takes into account the effect of compressible fluid velocity. It is also shown that, contrary to the single-charge model A, there are several correction exponents in the effective model, which are calculated at the leading order of the ɛ= 4 − d expansion.
We study a dilute and ultracold Bose gas of interacting atoms by using an effective field theory which takes into account the finite-range effects of the interatomic potential. Within the formalism of functional integration from the grand canonical partition function, we derive beyond-mean-field analytical results which depend on both the scattering length and the effective range of the interaction. In particular, we calculate the equation of state of the bosonic system as a function of these interaction parameters both at zero and finite temperature including one-loop Gaussian fluctuation. In the case of zero-range effective interaction, we explicitly show that, due to quantum fluctuations, the bosonic system is thermodynamically stable only for very small values of the gas parameter. We find that a positive effective range above a critical threshold is necessary to remove the thermodynamical instability of the uniform configuration. Remarkably, also for relatively large values of the gas parameter, our finite-range results are in quite good agreement with recent zero-temperature Monte Carlo calculations obtained with hard-sphere bosons.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by the latest high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green's function approach allows for calculating the excitation spectrum at finite temperature and for determining the effective masses of particles and holes.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green’s function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.
The grand canonical density functional theory for inhomogeneous systems of interacting bosons is developed in the effective action approach. The Legendre transform of the generating functional for Green's functions is used to define the effective action as a functional of both the particle density and the order parameter. Expanding the thermal effective action in powers of the Planck constant we obtain a systematic approximation scheme, which practically implements the Kohn-Sham idea: the problem of interacting bosons is reduced to a single-particle system in a fictitious external potential. The Kohn-Sham potential, the density and the order parameter have to be determined self-consistently in a given order approximation.