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Journal of Mathematical Physics

On the Theory of Quantum Corrections to the Equations of State and to the Particle-Distribution Functions

Gerard Nienhuis

Published Jan 1, 1970• 51 citations
Publisher

Abstract

The first-order quantum corrections to the equations of state of an almost-classical N-particle system are calculated to all orders in the particle density by expanding the normalized Wigner distribution function in powers of h2. In this way one avoids the expansion of the partition function, which has the unsatisfactory property that the correction terms diverge in the thermodynamic limit. Similarly, the first-order quantum correction to the pair distribution function is derived.

DOI
10.1063/1.1665053

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Journal of Mathematical Physics•Mar 1, 1970•0 citations
The first-order quantum corrections to the equations of state of an almost-classical N-particle system are calculated to all orders in the particle density by expanding the normalized Wigner distribution function in powers of h2. In this way one avoids the expansion of the partition function, which has the unsatisfactory property that the correction terms diverge in the thermodynamic limit. Similarly, the first-order quantum correction to the pair distribution function is derived.
We present a functional-integral method that is suitable for obtaining nonperturbative solutions to the partition function of fermion systems. The density matrix is cast as an expansion about the classical path contributions, and the quantum corrections can be calculated systematically by using retarded propagators in the space holomorphic functions. It is shown that the semiclassical expansion amounts to a high-temperature expansion, where the first nonvanishing quantum correction is of O(${\mathrm{\ensuremath{\beta}}}^{2}$). Each term of the semiclassical expansion may contain the coupling parameter in all orders of perturbation theory. The method can also be extended to boson systems.
The quantum corrections to the two-dimensional Wigner crystal, for filling \ensuremath{\nu}\ensuremath{\le}1/3, are discussed by using a Hartree-Fock expansion based on wave functions which are (i) related to one another by magnetic translations, (ii) orthonormal, and (iii) strongly localized. Such wave functions are constructed in terms of Gaussians that are localized at the sites of a triangular (Wigner) lattice and have a small overlap c. The ground-state energy per particle is calculated by an expansion in \ensuremath{\surd}\ensuremath{\nu} and in \ensuremath{\delta}\ensuremath{\equiv}c1/4{\mathit{c}}^{1/4}c1/4, which is rapidly convergent and stable under the thermodynamical limit. In particular, in this limit the cancellation of the infrared divergences occur order by order in the above expansion. The accurate control on the approximations allows a clear-cut comparison with the energy obtained by the Laughlin ansatz on the ground state and the numerical results confirm that the Wigner-crystal picture is energetically favored with respect to the Laughlin state for \ensuremath{\nu}1/9.
The quantum correction to the Vlasov equation is considered using the Wigner transform of the time-dependent Hartree-Fock equation without truncation in \ensuremath{\Elzxh} order. Decomposing the Wigner function in terms of test particles with the form factor in a phase space, quantum extension of the Vlasov equation determines the equation of motion of the test particle which follows classical dynamics. The discrimination between the quantum and classical limits is related to a density matrix condition which constrains the distribution of test particles.
A general nonperturbation problem, described by the Hamiltonian H\mathcal{H}H, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's T4{T}^{4}T4 correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for 〈Sz〉〈{S}^{z}〉〈Sz〉, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.
The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.
The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.