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arXiv:2109.00349 · Physical Review A

Path-integral approach to the thermodynamics of bosons with memory: Partition function and specific heat

T. Ichmoukhamedov, Jacques Tempere

Published Aug 25, 2021• 5 citations
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Abstract

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.

DOI
10.1103/physreva.104.023322

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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.
We set up a recursion relation for the partition function of a fixed number of harmonically confined bosons. For an ideal Bose gas this leads to the wellknown results for the temperature dependence of the specific heat and the ground-state occupancy. Due to the diluteness of the gas, we include both the isotropic contact interaction and the anisotropic dipole-dipole interaction by an infinite-bubble sum of the lowest-order perturbative results. Due to the anisotropy of the dipole-dipole interaction, the thermodynamic quantities of interest crucially depend on the trap configuration. Keywords: Dipolar gas; Bose-Einstein condensation; Canonical ensemble. 1. Many-Body Path Integral Formalism The aim of this work is to give a path-integral description for a system of a fixed particle number N. To this end, we start with the imaginary-time evolution amplitude which is given by the following N-fold path integral (x1,..., xN; τb | x ′ 1,..., x ′ N; τa) B ≡ 1 ∑ N∏ [ ∫ xn(τb)=xP(n) N!
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.