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Physical Review Letters

Exactly Soluble Quantum-Statistical System

R. G. Storer

Published Jan 5, 1970• 12 citations
Publisher

Abstract

The partition function for a system of bosons or fermions interacting via pairwise harmonic forces is evaluated exactly. It is found that the partition function has a structure similar to that of a system of independent bosons or fermions in an external harmonic well with, however, an important modification which implies a greatly reduced density of states.

DOI
10.1103/physrevlett.24.5

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The partition function for a system of bosons or fermions interacting via pairwise harmonic forces is evaluated exactly. It is found that the partition function has a structure similar to that of a system of independent bosons or fermions in an external harmonic well with, however, an important modification which implies a greatly reduced density of states.
Exact solutions to many-body interacting systems of both bosonic and fermionic particles confined to harmonic potential in an arbitrary number of dimensions are given. Energy levels and their degeneracies for trapped identical particles interacting via harmonic forces are calculated. This specific form of the interaction allows for analytical solutions. The mutual interaction, attractive or repulsive, modifies significantly the properties of the considered system. For a large number of particles the interaction essentially results in a frequency shift. Statistical properties (e.g., microcanonical and grand canonical partition functions) as well as some illustrative, physically relevant examples are discussed. Our results give an unusual opportunity for further studies of interacting systems in the framework of the exactly soluble model.
The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.
Coherent states are introduced as test functions to formulate the statistical mechanics of fermions and bosons interacting via schematic forces. Finite temperature solutions to the Lipkin model and to the Sch\"utte--Da Providencia model are obtained by performing the statistical sum aˋla\`a\mathrm{la}aˋla Hecht, e.g., by using coherent states. Comparison between present and exacts results is discussed.
The partition function, a fundamental tenet in statistical thermodynamics, contains in principle all thermodynamic information about a system. It encapsulates both microscopic information through the quantum energy levels and statistical information from the partitioning of the particles among the available energy levels. For identical particles, this statistical accounting is complicated by the symmetry requirements of the allowed quantum states. In particular, for Fermi systems, the enforcement of the Pauli principle is typically a numerically demanding task, responsible for much of the cost of the calculations. The interplay of these three elements---the structure of the many-body spectrum, the statistical partitioning of the NNN particles among the available levels, and the enforcement of the Pauli principle---drives the behavior of mesoscopic and macroscopic Fermi systems. In this paper, we develop an approach for the determination of the partition function, a numerically difficult task, for systems of strongly interacting identical fermions and apply it to a model system of harmonically confined, harmonically interacting fermions. This approach uses a recently introduced many-body method that is an extension of the symmetry-invariant perturbation method (SPT) originally developed for bosons. It uses group theory and graphical techniques to avoid the heavy computational demands of conventional many-body methods which typically scale exponentially with the number of particles. The SPT application of the Pauli principle is trivial to implement since it is done ``on paper'' by imposing restrictions on the normal-mode quantum numbers at first order in the perturbation. The method is applied through first order and represents an extension of the SPT method to excited states. Our method of determining the partition function and various thermodynamic quantities is accurate and efficient and has the potential to yield interesting insight into the role played by the Pauli principle and the influence of large degeneracies on the emergence of the thermodynamic behavior of large-NNN systems.
We describe a method to compute thermodynamic quantities in the harmonic approximation for identical bosons and fermions in an external confining field. We use the canonical partition function where only energies and their degeneracies enter. The number of states of given energy and symmetry is found by separating the center-of-mass motion, and by counting the remaining states of given symmetry and excitation energy of the relative motion. The oscillator frequencies that enter the harmonic Hamiltonian can be derived from realistic model parameters, and the method corresponds to an effective interaction approach based on harmonic interactions. To demonstrate the method, we apply it to systems in two dimensions. Numerical calculations are compared to a brute force method, which is considerably more computationally intensive.