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arXiv:2109.00350 · Physical Review A

Path-integral approach to the thermodynamics of bosons with memory: Density and correlation functions

T. Ichmoukhamedov, Jacques Tempere

Published Dec 1, 2021• 1 citation
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Abstract

Expanding upon previous work, using the path-integral formalism we derive expressions for the one-particle reduced density matrix and the two-point correlation function for a quadratic system of bosons that interact through a general class of memory kernels. The results are applied to study the density, condensate fraction, and pair correlation function of trapped bosons harmonically coupled to external distinguishable masses.

DOI
10.1103/physreva.104.062201

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Expanding upon previous work, using the path-integral formalism we derive expressions for the one-particle reduced density matrix and the two-point correlation function for a quadratic system of bosons that interact through a general class of memory kernels. The results are applied to study the density, condensate fraction, and pair correlation function of trapped bosons harmonically coupled to external distinguishable masses.
For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses.
A generalization of symmetrized density matrices in combination with the technique of generating functions allows one to calculate the partition function of identical particles in a parabolic confining well. Harmonic two-body interactions (repulsive or attractive) are taken into account. Also the influence of a homogeneous magnetic field, introducing anisotropy in the model, is examined. Although the theory is developed for fermions and bosons, special attention is paid to the thermodynamic properties of bosons and their condensation.
The dynamics of a one-dimensional quantum system coupled to a harmonic bath can be expressed through Feynman’s path integral expression for the reduced density matrix. In this expression the influence of the environment is seen in correlations between positions of the system that are nonlocal in time. Makri and Makarov [J. Chem. Phys. 102, 4600 (1995)] showed that for many practical problems correlations over only a few time steps, Δkmax, need to be taken into account, which led to an efficient iterative scheme. However, this algorithm scales as the size of the system to the power of 2(Δkmax+1), which restricts the size of the system that can be studied with this method. In this work we present an efficient algorithm which scales linearly with Δkmax. In our method the reduced density matrix is written as a convolution of its past values with an integral equation kernel. The calculation of that kernel is based on a perturbative expansion of the discretized quasiadiabatic path integral expression for the reduced density matrix. The expansion ignores certain types of correlations.
The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of N N harmonically trapped bosons interacting via contact interaction, where N N ranges from 2 2 to 10^4 10 4 . The ground-state energies are calculated, and compared to existing methods, including the analytic case (for N=2) N = 2 ) and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties such as the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide. This collectively demonstrates the capacity of the reduced density matrix method to accurately calculate collective ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.
Two basic correlation functions are calculated for a model of N harmonically interacting identical particles in a parabolic potential well. The density and the pair correlation function of the model are investigated for the boson case. The dependence of these static response properties on the complete range of the temperature and of the number of particles is obtained. The calculation technique is based on the path integral approach of symmetrized density matrices for identical particles in a parabolic confining well.
A path-integral representation for the kernel of the evolution operator of general Hamiltonian systems is reviewed. We study the models with bosonic and fermionic degrees of freedom. A general scheme for introducing boundary conditions in the path-integral is given. We calculate the path-integral for the systems with quadratic first class constraints and present an explicit formula for the heat kernel in this case. These results may be applied to many quantum systems which can be reduced to the Hamiltonian systems with quadratic constraints (confined quarks, Calogero type models, string and ppp-brain theories, etc.).