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Journal of Mathematical Sciences

Calculation of the asymptotics of the two-point correlation function for one-dimensional Bose gas

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published May 1, 2008• 0 citations
Publisher
DOI
10.1007/s10958-008-9001-y

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A quantum field-theoretical model which describes a spatially nonhomogeneous one-dimensional nonrelativistic repulsive Bose gas in an external harmonic potential is considered. The two-point thermal correlation function of the Bose gas is calculated in the framework of the functional integration approach. The calculations are done in the coordinate representation. The method of successive integration first over the “high-energy” functional variables and then over the “low-energy” ones is used. The effective action functional for the low-energy variables is calculated in one-loop approximation. The functional integral representation for the correlation function is obtained in terms of the low-energy variables and is estimated with the help of stationary phase approximation. The asymptotics of the correlation function is studied in the limit when the temperature tends to zero while the volume occupied by the nonhomogeneous Bose gas increases infinitely. It is demonstrated that the behavior of the thermal correlation function in this limit is power-like and is governed by the critical exponent that depends on the spatial and thermal arguments.
A quantum field-theoretical model, which describes spatially non-homogeneous repulsive Bose gas in an external harmonic potential is considered. Two-point thermal correlation functions of the Bose gas are calculated in the framework of the functional integration approach. Successive integration over the ``high-energy'' functional variables first and then over the ``low-energy'' ones is used. The effective action functional for the low-energy variables is obtained in one loop approximation. The functional integral representations for the correlation functions are estimated by means of the stationary phase approximation. A power-law asymptotical behaviour of the correlators of the one-dimensional Bose gas is demonstrated in the limit, when the temperature is going to zero, while the volume occupied by the non-homogeneous Bose gas infinitely increases. The power-law behaviour is governed by the critical exponent dependent on the spatial arguments.