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arXiv:cond-mat/0604413 · St Petersburg Mathematical Journal

Functional integration and the two-point correlation function of the one-dimensional Bose gas in a harmonic potential

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published Jan 19, 2006• 2 citations
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Abstract

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.

DOI
10.1090/s1061-0022-06-00893-4

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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.
Abstract We present a comprehensive study shedding light on how thermal fluctuations affect correlations in a Bose gas with contact repulsive interactions in one spatial dimension. The pair correlation function, the static structure factor, and the one-body density matrix are calculated as a function of the interaction strength and temperature with the exact ab-initio Path Integral Monte Carlo method. We explore all possible gas regimes from weak to strong interactions and from low to high temperatures. We provide a detailed comparison with a number of theories, such as perturbative (Bogoliubov and decoherent classical), effective (Luttinger liquid) and exact (ground-state and thermal Bethe Ansatz) ones. Our Monte Carlo results exhibit an excellent agreement with the tractable limits and provide a fundamental benchmark for future observations which can be achieved in atomic gases, cavity quantum-electrodynamic and superconducting-circuit platforms.
The behavior of the spatial two-particle correlation function is surveyed in detail for a uniform one-dimensional Bose gas with repulsive contact interactions at finite temperatures. Long-, medium-, and short-range effects are investigated. The results span the entire range of physical regimes from ideal gas to strongly interacting and from zero temperature to high temperature (Gross-Pitaevskii) and strongly interacting (Tonks-Girardeau) gases. We present perturbative analytic methods, available at strong and weak couplings, and first-principles numerical results using imaginary time simulations with the gauge-PPP representation in regimes where perturbative methods are invalid. Nontrivial effects are observed from the interplay of thermally induced bunching behavior versus interaction induced antibunching.
Introduction In the previous chapters correlation functions were represented as determinants of a special form of Fredholm integral operator. First, a differential equation was obtained for the time-independent zero temperature field correlation function for the impenetrable Bose gas (it depends only on one variable—distance). By means of an isomonodromic deformation technique A. Jimbo, T. Miwa, Y. Mori and M. Sato wrote down the ordinary differential equation (of Painlevè type). In the previous chapter differential equations for correlation functions in more general situations (with time, temperature and for finite coupling) were constructed. A.R. Its came up with the idea of applying the matrix Riemann problem for the description of these correlation functions (Fredholm determinants). This permits us to write down partial differential equations (completely integrable) for correlation functions and evaluate the asymptotics of correlation functions. Below we follow this line. Integrable nonlinear partial differential equations for correlation functions of the one-dimensional nonrelativistic Bose gas with point-like repulsion between gas particles were given in section XIV. So quantum correlation functions can be expressed in terms of the solutions of classical nonlinear integrable evolution systems. These classical systems have been investigated already by means of the classical inverse scattering method. Our approach allows one, in particular, to solve the particularly interesting problem of calculating the asymptotics of correlators at large time and distance; this is done in the next section.
We investigate the behavior of the anomalous correlation function in two-dimensional Bose gas. In the local case, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities are also considered. These effects can modify, in particular, the chemical potential, the ground-state energy, the depletion, and the superfluid fraction. Our predictions are in good agreement with recent analytical and numerical calculations. We show also that the anomalous density presents a significant importance compared to the noncondensed one at zero temperature. The single-particle anomalous correlation function is expressed in two-dimensional homogenous Bose gases by using the density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous manner the true condensate at zero temperature, while it does not exist at finite temperature.